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# A chimera model for motion anticipation in the retina and the primary visual cortex

Jérôme Emonet, Selma Souihel, Matteo Di Volo, Alain Destexhe, Frédéric

Chavane, Bruno Cessac

► To cite this version:

Jérôme Emonet, Selma Souihel, Matteo Di Volo, Alain Destexhe, Frédéric Chavane, et al. A chimera model for motion anticipation in the retina and the primary visual cortex. Neural Computation, 2025, 37 (11), pp.1925-1974. 10.1162/neco.a.34. hal-04709925v2

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Submitted on 11 Apr 2025

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Distributed under a Creative Commons Attribution 4.0 International License
A CHIMERA MODEL FOR MOTION ANTICIPATION
IN THE RETINA AND THE PRIMARY VISUAL
CORTEX

**Jerome Emonet**

Université Côte d'Azur, Inria
Biovision team and Neuromod Institute
Sophia Antipolis, France
jerome.emonet@inria.fr

**Selma Souihel**

P16 - Programme IA
Inria Rocquencourt
Domaine de Voluceau
78150 Le Chesnay-Rocquencourt
selma.souihel@inria.fr

**Frederic Chavane**

Institut de Neurosciences de la Timone UMR7289
CNRS Aix-Marseille University
13385 Marseille Cedex 05, France
frederic.chavane@univ-amu.fr

**Alain Destexhe**

Institut NeuroPSI - UMR9197
CNRS Paris-Saclay University
Campus CEA Saclay, Saclay, FRANCE
Alain.Destexhe@cnrs.fr

**Matteo di Volo**

Università Claude Bernard Lyon 1
Institut National de la Santé et de la Recherche Médicale,
Stem Cell and Brain Research Institute U1208
Bron, France
matteo.di-volo@univ-lyon1.fr

**Bruno Cessac**

Université Côte d'Azur, Inria
Biovision team and Neuromod Institute
Sophia Antipolis, France
bruno.cessac@inria.fr

April 10, 2025
**ABSTRACT**

We propose a mean field model of the primary visual cortex (V1), connected to a realistic retina model, to study the impact of the retina on motion anticipation. We first consider the case where the retina does not itself provide anticipation - which is then only triggered by a cortical mechanism, the "anticipation by latency" - and unravel the effects of the retinal input amplitude, of stimulus features such as speed and contrast, and of the size of cortical extensions and fibre conduction speed. Then, we explore the changes in the cortical wave of anticipation when V1 is triggered by retina-driven anticipatory mechanisms: gain control and lateral inhibition by amacrine cells. Here, we show how retinal and cortical anticipation combine to provide an efficient processing where the simulated cortical response is in advance over the moving object that triggers this response, compensating the delays in visual processing.

**Keywords:** Visual processing · Anticipation · Retina · V1 · Lateral connectivity

# 1 Introduction

The brain is not just a reactive encoder, it is able to respond in a proactive way, especially with regards to stimuli. In visual processing, it is able to extrapolate and estimate how the visual stimulus is the most likely to behave in a near future, given the information of the past. This is, of course, based on the assumption that the stimulus is predictable to some extent. In particular, several studies have suggested that the predictability of stimuli can be learned from spatial and temporal regularities, arising e.g. in a deterministic trajectory. However, one can ask at which stage of the visual system do these predictions start taking place. At the level of the early stages of the visual system ? Or in late-stage processors ?

Prediction is a rather wide concept. If, by this term, one refers to motion extrapolation, studies have reported that it already starts at the level of the retina. Berry et al. (1999) have first shown that local gain control mechanisms, occurring at the level of retinal bipolar and ganglion cells, can explain a form of local anticipation for a moving bar by advancing the peak in the retinal ganglion cells response. This explains the change in the shape of responses observed in experimental data - bringing the ganglion cells to their activity peak earlier than when they respond to a flashed bar - without modifying the time at which their activity starts increasing, i.e. when the bar enters in their receptive field. Another study by Johnston and Lagnado (2015) has emphasized the role of lateral inhibition in the elicitation of anticipatory mechanisms at the retinal level. They have proposed that motion anticipation can be mediated via feedforward inhibition from amacrine cells inputs that specifically suppress the response to the moving object in the latter half of the receptive field. This truncates the response and yields an early response peak. In this case, motion anticipation arises from the general properties of the retina connectome and from the feedforward inhibition that ganglion cells receive from amacrine cells. Souihel and Cessac (2021) proposed, in a modelling study, that feedback inhibition due to amacrine cells induces a wave of activity further anticipating the bipolar cell response and thereby enhancing the effect of gain control. Finally, Menz et al. (2020) observed that motion anticipation can, in addition to a peak advancement
by truncation of the response, arise by advancing the *onset* of the response, with a strong
anticipatory effect.

It has also been shown that anticipation is further carried out at the level of the primary visual cortex. Jancke et al. (2004) first demonstrated the existence of anticipatory mechanisms in the cat primary visual cortex. They recorded cells in the central visual field of area 17 of anesthetized cats, responding to small squares of light, either flashed or moving in different directions, and with different speeds. When presented with the moving stimulus, these cells show a reduction of neural latencies, as compared to the flashed stimulus. Guo et al. (2007) noted that a spatio-temporal sequence of bars, of the same orientation, flashed at the periphery of the receptive field of a V1 neuron, generates an increase in the neuron's activity if the orientation matches its preferred orientation. This phenomenon does not occur if the spatiotemporal sequence is randomised. Subramaniyan et al. (2018) have reported the existence of similar anticipatory effects in the macaque primary visual cortex, showing that a moving bar is processed faster than a flashed bar. These observations are in favor of a collaborative work conducted, on the one hand, by the retina and V1 to help reducing the latencies, and, on the other hand, by other specialized brain regions which carry out predictive computations. Learning and training seem also to play important roles in calibrating the response of the nervous system to a giving moving object. Finally, Benvenuti et al. (2015, 2020) have studied the trajectory-based activity in V1. They have compared voltage sensitive dye imaging responses (VSDI) for different cortical locations along a bar trajectory. By centering all positions on the same relative time events (bar centered in the middle of the receptive field), they have highlighted a gradient in the response (see also Fig. 2). The further the bar starts from the current position of a cortical column the earlier its activity rises. They gave convincing arguments that this increase is carried by the horizontal (intra-cortical) connectivity in the cortex.

At the current state of the art there are thus convincing benches of evidence that antic-
ipation takes place along the pathway from retina to cortex. This has been observed
with a large diversity of recording conditions (in vitro, in vivo, anesthetized, awake), for
different animal models (rodent, cat, monkey ... ). This suggests that there might exist a
synergy between the retina, the thalamus-LGN and the cortex to optimize anticipation.
This leads to a natural question: How does the dynamical response of the retina to a
moving object affect the dynamics changes in the cortex ?

Along these lines, the purpose of this paper is to understand what the retina brings to the
primary visual cortex in terms of anticipation, using a computational model. Rather than
developing a biologically plausible model for one species, the aim is to reconcile two
sets of work conducted to model different species, and to study the potential effect of
known retinal anticipatory mechanisms on cortical anticipation. We have first of a retinal
model, developed in (Souihel and Cessac, 2021; Cessac, 2022; Kartsaki et al., 2024) to
confront several potential mechanisms of retinal anticipation (gain control, amacrine
cells lateral inhibition). This is the input of a V1 mean-field model developed in Zerlaut
et al. (2018); Chemla et al. (2019), and calibrated on experimental VSDI signal data
on the monkey cortex. We can then tune this model to compare cortical anticipation in
the absence or presence of retinal anticipation. We do not consider the effects of the
thalamus (LGN) in this work, which transparent, treated as a simple relay.

The paper is organized as follows. Section 2 is devoted to methods. We introduce the
retino-cortical model as well as specific quantities (observables) used throughout the
paper to quantify anticipation. Section 3 presents our main results. In section 3.1 we
study the simulated VSDI signal response to a moving bar when the retina is passive and
brings no anticipatory effect (control conditions). We show that the model qualitatively
reproduces the results experimentally observed in (Benvenuti et al., 2020) with, however,
quantitative deviations, due to the lack of retinal anticipation mechanisms. We analyse, in
these control conditions, the effect of varying the retinal output amplitude (section 3.2.1)
and show that it impacts the anticipation, even if the retina is passive. We also investigate
the role of the stimulus speed (section 3.2.2) Here, we point out that the behaviour of the
anticipation time as a function of the bar speed is in contradiction with what is observed
experimentally in Berry et al. (1999); Johnston and Lagnado (2015). As we show, this is
again due to the lack of active retinal anticipation mechanisms. Finally, still in control
conditions, we study the effect of cortical extent (section 3.2.3), and the conduction
speed in cortical fibres (section 3.2.4). In section 3.3 we analyse the effect of retinal
anticipation mechanisms on cortical anticipation. The section 3.3.1 considers the impact
of gain control applied either to bipolar cells or retinal ganglion cells. In section 3.3.2
we compare the influence of two types of retinal connectivities on cortical anticipation:
1) feedforward connection from bipolar cells to amacrine cells to retinal ganglion cells;
2), feedback, reciprocal, connections between bipolar and amacrine cells, both imputing
retinal ganglion cells. In section 4 we discuss several potential extensions of this work
such as more complete models of the retina or the inclusion of a mean-field model of
the thalamus. The appendix contains a table with the model's parameters and a table
synthesizing the effects of active retinal mechanisms on each anticipation observable
and comparing them to experimental and control conditions. The Supplementary section
contains additional results to complete the main text.

**2 Methods**

**2.1 The retina model**

The model is made of 3 bidimensional layers whose structure is shown in Fig. 1: A layer
of bipolar cells (BCs), a layer of amacrine cells (ACs) and a layer of retinal ganglion
cells (RGCs). The layers are rectangle grids, with the same dimension $L_x \times L_y$ mm$^2$
and the same number of cells, *N*. BCs are labelled with an index *i* = 1... *N*, ACs are
labelled with an index *j* = 1... *N*, RGCs with an index *k* = 1... *N*. Cells are located
on the nodes of their grid layer and are spaced by a distance *δ* mm in both directions *x*, *y*.
Spatial coordinates are noted $\vec{x} = (x, y)$ and the coordinates of, e.g., BC *i* are $(x_i, y_i)$.

The 3 layers ought to be parametrised by a vertical coordinate, *z*. However, this parametri-
sation is implicit in the retinal layers label BCs, ACs, RGCs and we are not going to use
a vertical distance between layers. Since all layers have the same grid spacing, *δ*, and
the same number of neurons, there is a vertical alignment of nodes: the BC with e.g.
index $i = 10$ is vertically aligned with the AC of index $j = 10$ and the RGC with index $k = 10$. This given, we define the Euclidean distance between a cell $l$, in a layer, and a cell $m$, in the same or another layer, as $d(l, m) = \sqrt{(x_l - x_m)^2 + (y_l - y_m)^2}$. That is, we do not consider the vertical distance, for simplicity. The distance $d(l, m)$ is used for connectivity patterns.

There is indeed an interlayer connectivity. The connectivity from BCs to ACs is characterized by a connectivity matrix $\Gamma_A^B$, with entries $\Gamma_{A_j}^{B_i} = 1$ if there is a connection from BC $i$ to AC $j$, and $\Gamma_{A_j}^{B_i} = 0$ otherwise. In the paper, the connectivity structure of $\Gamma_A^B$ is called "nearest-neighbours+1": the BC with coordinate $i$ connects to the AC with coordinate $j$ and with the four nearest neighbours of this AC. The synaptic weight from BC $i$ to AC $j$ is then $W_{A_j}^{B_i} = w_A^B \Gamma_{A_j}^{B_i}$ where the parameter $w_A^B \ge 0$ controls the excitatory synapses amplitude. This form of synaptic weight allows us to tune the synaptic intensity from BCs to ACs with the unique parameter $w_A^B$. The connectivity from ACs to BCs is also characterized by a synaptic weight matrix $W_B^A$ with entries $W_{B_i}^{A_j} = w_B^A \Gamma_{B_i}^{A_j}$, with $w_B^A \le 0$ (inhibition from AC $j$ to BC $i$), where $\Gamma_B^A$ is "one to one" (AC $j$ only connects to BC with index $i = j$). The synapse from BC $i$ to RGC $k$ corresponds to "Gaussian pooling" (Berry et al., 1999) where $W_{G_k}^{B_i} = w_G^B \frac{e^{-\frac{d(i,k)^2}{2\sigma^2}}}{2\pi\sigma^2}$ with $w_G^B \ge 0$. Likewise, the connectivity from AC $j$ to RGC $k$ is characterized by a synaptic weight $W_{G_k}^{A_j} = w_G^A \frac{e^{-\frac{d(j,k)^2}{2\sigma^2}}}{2\pi\sigma^2}$ with $w_G^A \le 0$. Synaptic weights are expressed in Hz.

Cell types have characteristic times, expressed in seconds, corresponding to the integration time of their response to external influence and including synaptic and transmission delay. Here, we consider that all BCs have the same characteristic time, $\tau_B$. Likewise, all ACs have a characteristic time, $\tau_A$, and RGCs a characteristic time, $\tau_G$.

The dynamics of cells is based on their voltage. We note $V_{B_i}$ the voltage of BC $i$ and so on. Voltage rectification takes place below a certain threshold (eq. (3)). In addition, BCs and RGCs have gain control, a desensitization when activated by a steady illumination. (eq. (5), (8)), characterized by an activation variable $A_{B_i}$ for BCs, $A_{G_k}$ for RGCs (Chen et al., 2013). The dynamics of voltages and activations is given by eq. (2) below.

BCs receive a visual input featuring the pre-processing of a visual stimulus via photo-receptors and horizontal cells. In this study, a visual stimulus is a grey scale video, that is a function $S(\vec{x}, t) \in [0, 1]$ where 0 corresponds to black and 1 to white. The pre-processing of this visual stimulus via photo-receptors and horizontal cells is modelled by a spatio-temporal convolution:

$$ \left[ K_{B_i}^{\vec{x},t} * S \right](t) \equiv V_{i_{\text{drive}}}(t), \qquad (1) $$

where $K_{B_i}(\vec{x}, t)$ is a spatio-temporal kernel, centered at the coordinate of the BC $i$, and called "OPL kernel" (where "OPL" stands for "Outer Plexiform Layer" https://www.ncbi.nlm.nih.gov/books/NBK11518/). Spatially, this is a Gaussian with a center of radius $\sigma_c$ and temporally, a gamma function with a characteristic time $\tau_c$ (see appendix 5.1 for the value of these parameters). Therefore, the OPL input of BCs is monophasic in space and time. The choice of a monophasic OPL input may appear
in contradiction with the literature. Indeed, the temporal receptive field of bipolar cells, measured by reverse correlation to white noise, is found to be biphasic (Baccus et al., 2008). Similar results are found in Strauss et al. (2022) where, measuring glutamate release, they conclude to the influence of amacrine cells in shaping this biphasic response. But, precisely, as shown in (Kartsaki et al., 2024) the presence of lateral inhibition by ACs in our model allows to generate biphasic profiles, in space and in time, for the BCs response even if the OPL input is monophasic. Note that the spatial RF is circular: we do not consider orientation selective cells in this paper. In the definition of $K_{B_i}$ there is a multiplicative factor, $C$, which allows us to control the amplitude of $V_{idrive}(t)$. This is used, in section 3.2.1, to modify the amplitude of the retinal input to the cortex.

The joint evolution of BCs, ACs, RGCs, driven by the stimulus $S$ is given by the following set of equations. We refer to the papers (Souihel and Cessac, 2021; Cessac, 2022; Kartsaki et al., 2024) for detail about this dynamical system and its mathematical study.

$$ 
\left\{ 
\begin{aligned}
\frac{dV_{B_i}}{dt} &= -\frac{V_{B_i}}{\tau_B} + \sum_{j=1}^{N_A} W_{B_i}^{A_j} V_{A_j} + V_{idrive}(t), \\
\frac{dA_{B_i}}{dt} &= -\frac{A_{B_i}}{\tau_{a_B}} + h_B \mathcal{N}_B(V_{B_i}), \\
\frac{dV_{A_j}}{dt} &= -\frac{V_{A_j}}{\tau_A} + \sum_{i=1}^{N_B} W_{A_j}^{B_i} R_B(V_{B_i}, A_{B_i}), \\
\frac{dV_{G_k}}{dt} &= -\frac{V_{G_k}}{\tau_G} + \sum_{j=1}^{N_A} W_{G_k}^{A_j} V_{A_j} + \sum_{i=1}^{N_B} W_{G_k}^{B_i} R_B(V_{B_i}, A_{B_i}), \\
\frac{dA_{G_k}}{dt} &= -\frac{A_{G_k}}{\tau_{a_G}} + h_G \mathcal{N}_G(V_{G_k}).
\end{aligned}
\right. 
\quad (2) $$

Here, in addition to voltages, we have introduced the activity variables, $A_{B_i}$ for BCs, $A_{G_k}$ for RGCs, ruling the gain control mechanisms on these cell type. There is no gain control on ACs. BCs are, in addition, rectified. The function:

$$ \mathcal{N}_B(V_{B_i}) = \begin{cases} 0, & \text{if } V_{B_i} \le \theta_B; \\ V_{B_i} - \theta_B, & \text{else,} \end{cases} \quad (3) $$

models this BCs voltage rectification with a threshold $\theta_B$. The BCs output to ACs and RGCs is then characterized by a non linear response to its voltage variation, given by:

$$ R_B(V_{B_i}, A_{B_i}) = \mathcal{N}_B(V_{B_i}) \mathcal{G}_B(A_{B_i}), \quad (4) $$

where the function:

$$ \mathcal{G}_B(A) = \begin{cases} 0, & \text{if } A \le 0; \\ \frac{1}{1+A^6}, & \text{otherwise.} \end{cases} \quad (5) $$

implement the gain control of BCs as a function the activity variable $A_{B_i}$ (Berry et al., 1999).

As ganglion cells are spiking cells, their response function is:

$$ R_G(V_{G_k}, A_{G_k}) = \mathcal{N}_G(V_{G_k}) \mathcal{G}_G(A_{G_k}). \quad (6) $$

This function corresponds to a probability of firing within a small time interval. A non-linearity is fixed so as to impose an upper limit over the firing rate:

$$ \mathcal{N}_G(V) = \begin{cases} 0, & \text{if } V \le 0; \\ \alpha_G(V - \theta_G), & \text{if } \theta_G \le V \le N_G^{\text{max}}/\alpha_G + \theta_G; \\ N_G^{\text{max}}, & \text{else.} \end{cases} \quad (7) $$
We have, for the RGCs gain control:

$$G_G(A) = \begin{cases} 0, & \text{if } A \le 0; \\ \frac{1}{1+A}, & \text{else.} \end{cases} \qquad (8)$$

which actually differs from the non-linearity in the BCs gain control, following (Chen et al., 2013). The gain control rate, $h_B$ for BCs, $h_G$ for RGCs, expressed in Hz mV$^{-1}$, tunes the intensity of the gain control. In particular, if $h_B = 0$, $A_{B_i} \to 0$ exponentially fast so that the gain $G_B(A) = 1$. The same remark holds for RGCs.

Parameters values of the model can be found in the appendix 5.1.

## 2.2 The cortical model

This is a two dimensional model composed of two populations of cortical columns: excitatory (E) and inhibitory (I), located in a cortical area of dimension $\alpha L_x \times \alpha L_y$, where $\alpha$ is a magnification factor from retina to cortex. Cortical columns represent the spatial average of cortical neurons at a space scale roughly corresponding to one pixel of voltage sensitive dye imaging (VSDI signal) (Chemla, 2010). This model has actually been employed in (Chemla et al., 2007) to reproduce the VSDI response to a simple visual stimuli (apparent motion) in the awake monkey primary visual cortex (V1). Cortical spatial coordinates are noted $(x, y) \equiv \vec{x}$. We thus use the same notations as for the retina, to alleviate notations, although there is a magnification factor between these two systems of coordinates.

The activity of cortical columns is represented by their average firing rate: $\nu_E$ for excitatory columns, $\nu_I$ for inhibitory columns. The equations for cortical neurons dynamics are based on a mean-field model of Adapting Exponential (AdEx) neurons (DiVolo et al., 2019). This model was derived under the hypothesis that the network dynamics is Markovian at a timescale of a few ms and stationary for a duration $T$. One describes then the collective dynamics through a master equation formalism developed by ElBoustani and Destexhe (2009). This system can reproduce asynchronous irregular regime, a typical feature of the awake states, as well as Up and Down states, characteristic of sleep or anesthesia states (DiVolo et al., 2019).

Here we extend the dynamic mean-field equations of (ElBoustani and Destexhe, 2009; Zerlaut et al., 2018; Chemla et al., 2019; DiVolo et al., 2019; Di Volo et al., 2019; DiVolo and Férézou, 2021) to:

$$\left\{ \begin{aligned} T \frac{\partial \nu_E(\vec{x},t)}{dt} &= -\nu_E(\vec{x},t) + F_E [\mathcal{I}_E^E(\vec{x},t), \mathcal{I}_E^I(\vec{x},t)] \\ T \frac{\partial \nu_I(\vec{x},t)}{dt} &= -\nu_I(\vec{x},t) + F_I [\mathcal{I}_I^E(\vec{x},t), \mathcal{I}_I^I(\vec{x},t)]. \end{aligned} \right. \qquad (9)$$

where:

$$\left\{ \begin{aligned} \mathcal{I}_E^E(\vec{x}, t) &= \nu^{aff}(\vec{x}, t) + \nu^{drive} + A_E^E \nu_E^{input}(\vec{x}, t), \\ \mathcal{I}_E^I(\vec{x}, t) &= A_E^I \nu_I^{input}(\vec{x}, t), \\ \mathcal{I}_I^E(\vec{x}, t) &= \nu^{aff}(\vec{x}, t) + \nu^{drive} + A_I^E \nu_E^{input}(\vec{x}, t), \\ \mathcal{I}_I^I(\vec{x}, t) &= A_I^I \nu_I^{input}(\vec{x}, t). \end{aligned} \right. \qquad (10)$$

Here, $\mathcal{I}_X^Y(\vec{x}, t)$, $X, Y = \{E, I\}^2$, is the total contribution of population Y controlling the time evolution of population X.
The term $\nu^{aff}(\vec{x}, t)$ in eq. (10) corresponds to the retino-thalamic input (sensory drive). As the thalamus is not considered here (we assimilate it to a simple relay) this input comes directly from the RGCs. There is a (linear) retinotopy between a point in the retina (RGC), and a point in V1 (cortical column) with a magnification factor $\alpha$ from the retina to V1. Each RGC inputs a cortical column and $\nu^{aff}(\vec{x}, t)$ is the firing rate emitted by the RGC that inputs the cortical column located at $\vec{x}$. In the absence of gain control or amacrine connectivity, the retinal model is reduced to a convolution cascade. In this case, the input $\nu^{aff}(\vec{x}, t)$ is therefore similar to the one used in (Benvenuti et al., 2020). In contrast to previous papers the retino-thalamic input $\nu^{aff}$ is, in our case, acting on the two populations, excitatory and inhibitory. In contrast to these papers too, we have introduced the coefficients $A_X^Y$ weighting the relative contributions of excitatory and inhibitory populations, $\nu_E^{input}$, $\nu_I^{input}$, from the other columns. This corresponds to the observed physiology of the real cortex. These coefficients were implicitly set to 1 in the previous papers.

In eq. (10) the terms $\nu_E^{input}(\vec{x}, t)$ (resp. $\nu_I^{input}(\vec{x}, t)$) are excitatory (resp. inhibitory) inputs coming from the column itself or from other cortical columns. They are given by:

$$ \left\{ \begin{aligned} \nu_E^{input}(\vec{x}, t) &= \int_{\mathbb{R}} \mathcal{N}_E(\vec{x} - \vec{x}') \nu_E(\vec{x}', t - \|\vec{x}' - \vec{x}\|/v_c) d\vec{x}' \\ \nu_I^{input}(\vec{x}, t) &= \int_{\mathbb{R}} \mathcal{N}_I(\vec{x} - \vec{x}') \nu_I(\vec{x}', t - \|\vec{x}' - \vec{x}\|/v_c) d\vec{x}' \end{aligned} \right. \quad (11) $$

where $\mathcal{N}_E, \mathcal{N}_I$ are 2D circular Gaussian connectivity kernels with mean-square deviation $\sigma_E, \sigma_I$:

$$ \mathcal{N}_X(\vec{x}) = \frac{e^{-\frac{1}{2} \frac{\|\vec{x}\|^2}{\sigma_X^2}}}{2\pi\sigma_X^2}, \quad (12) $$

with $X = E, I$. This form follows e.g. (Zerlaut et al., 2018), although our connectivity kernel is two dimensional in contrast to their paper. Thus, $\sigma_E, \sigma_I$ control the two dimensional cortical extent of the excitatory and inhibitory connections. Note, that, due to the normalisation of the Gaussian, the shorter the cortical extent the larger the amplitude of the synaptic weight $\mathcal{N}_X(\vec{x})$ (see section 3.2.3 for a consequence of this). This normalisation means that the number of connections within the system remains constant when the size of the extensions is increased. The parameter $v_c$ is the speed of axonal conduction (assumed to be a constant). Equations (11) therefore express that the excitatory input $\nu_E^{input}(\vec{x}, t)$ is the sum of the incoming excitatory activity from the connected columns arising with a delay $\|\vec{x}' - \vec{x}\|/v_c$ depending on the distance between the columns and the axonal conduction speed.

To go further in the explanation of the mean-field cortical equations we define, dropping the $(\vec{x}, t)$ dependence for legibility:

$$ \mu_{G_X^Y} = \mathcal{I}_X^Y K_X \tau_X Q_X \quad (13) $$

$$ \sigma_{G_X^Y} = Q_X \sqrt{\frac{\mathcal{I}_X^Y K_E \tau_X}{2}}; \quad X, Y = \{E, I\}^2. \quad (14) $$

where, $\mu_{G_X^Y}$ is contribution of population Y to the mean conductance of population X at $(\vec{x}, t)$, and $\sigma_{G_X}$ is the corresponding standard deviation, computed from shot-noise theory (Papoulis, 1965). $K_E$ (resp. $K_I$) is the number of excitatory synapses (resp. inhibitory),
$Q_E$ (resp. $Q_I$) the unitary excitatory conductance (resp. inhibitory), and $\tau_E$ (resp. $\tau_I$) the excitatory decay (resp. inhibitory). See table in appendix 5.1 for the value of these parameters.

The transfer functions $F_E$ (resp. $F_I$) of excitatory (resp. inhibitory) neurons then take the form:

$$F_X(\mathcal{I}_E^X, \mathcal{I}_X^I) = \frac{1}{2\tau_{V,X}} \operatorname{erfc}(\frac{\nu_{thr,X}^{\mathrm{eff}} - \mu_{V,X}}{\sqrt{2}\sigma_{V,X}}), \quad X = E, I, \qquad (15)$$

where:

$$\mu_{V,X} = \frac{\sum_{Y=E,I} \mu_{G_X^Y} V_Y + g_L V_L}{\mu_G}, \quad X = E, I, \qquad (16)$$

is the mean-voltage of the population $X$ in the cortical column, whereas $V_E, V_I, V_L$ are respectively the reversal potentials for excitatory (E), inhibitory (I) populations and for the leak. Likewise we have:

$$\sigma_{V,X} = \sqrt{\sum_{Y=E,I} K_Y \mathcal{I}_X^Y \frac{(U_Y \tau_Y)^2}{2(\tau_m^{\mathrm{eff}} + \tau_Y)}}, \qquad (17)$$

$$\tau_{V,X} = \frac{\sum_{Y=E,I} (K_Y \mathcal{I}_X^Y (U_Y \tau_Y)^2)}{\sum_{Y=E,I} (K_Y \mathcal{I}_X^Y (U_Y \tau_Y)^2 / (\tau_m^{\mathrm{eff}} + \tau_Y))}, \qquad (18)$$

where we defined $U_Y = \frac{Q_Y}{\mu_G}(V_Y - \mu_{V,Y})$ and where $X = (E, I)$. Note that the quantities defined in (16),(17),(18) depend non linearly on the dynamical variables (10).

The total input conductance of the neuron $\mu_G$ and its effective membrane time constant $\tau_m^{\mathrm{eff}}$ are controlled by the mean conductances as follows:

$$\mu_G = \sum_{X,Y \in \{E,I\}^2} \mu_{G_X^Y} + g_L, \qquad (19)$$

$$\tau_m^{\mathrm{eff}} = \frac{c_m}{\mu_G},$$

where $C_m$ is the membrane capacitance, assumed to be the same for all neurons and $g_L$ is the leak conductance.

Finally, in eq. (15), $\nu_{thr,X}^{\mathrm{eff}}$ is a phenomenological threshold expressed as a first order expansion of the three sub-threshold statistical quantities : $\mu_{V,X}$, $\sigma_{V,X}$, $\tau_{V,X}$. As it does not exist an analytic formulation of the transfer function for complex models such as the Adaptive Exponential IF or Hodgkin Huxley, Zerlaut et al. (2018) have developed a semi-analytic method based on a phenomenological threshold which carries the single neuron non-linearities mechanisms (spike/reset and adaptation). This leads to the following expression for the phenomenological threshold:

$$\nu_{thr,X}^{\mathrm{eff}} = P_{X,0} + \sum_{u \in \{\mu_{V,X}, \sigma_{V,X}, \tau_{V,X}^N\}} P_{X,u} \cdot \left(\frac{u - u^0}{\delta u^0}\right) + \sum_{u,v \in \{\mu_{V,X}, \sigma_{V,X}, \tau_{V,X}^N\}^2} P_{X,uv} \cdot \left(\frac{u - u^0}{\delta u^0}\right) \left(\frac{v - v^0}{\delta v^0}\right), \qquad (20)$$
where $\tau_{V,X}^N = \tau_{V,X}G_l/c_m$ is an unit-less parameter. Coefficients $P_{X,u}$ have been fitted on numerical simulations of a given single AdEx neuron model with conductance-based exponential synapses (Carlu et al., 2020). Note that they are different for excitators and inhibitors. The value of these coefficients in our model are given in the table of appendix 5.1, as well as the coefficients $\mu_V^0, \delta\mu_V^0, \sigma_V^0, \delta\sigma_V^0, (\tau_V^N)^0, (\delta\tau_V^N)^0$ which are actually assumed to be constant over the populations $E, I$.

The cortical model is designed to reproduce the pixels intensity of VSDI, corresponding to a variation of fluorescent luminosity with respect of the fluorescent baseline. Its expression depends on the average membrane potential of the excitatory and inhibitory populations in the column located at $\vec{x}$, at time $t$. Thus, the simulated VSDI signal is:

$$VSDI(\vec{x}, t) = 0.8 \times VSDI_E(\vec{x}, t) + 0.2 \times VSDI_I(\vec{x}, t), \quad (21)$$

where the coefficient 0.8 and 0.2 corresponds to the fraction of excitators and inhibitors in the column population, whereas:

$$VSDI_X(\vec{x}, t) = \frac{\mu_{V,X}(\vec{x}, t) - \mu_{V,X_0}(\vec{x})}{\mu_{V,X_0}(\vec{x})}, \quad X = E, I,$$

**Remark.** Note that this type of cortical model can exhibit pathological (i.e. model induced) oscillations due to bifurcations when some parameters become too large (such as $\nu^{drive}$ or the conduction speed). This phenomenon is well known and has been reported in the literature (Veltz, 2011). We observe as well such oscillations when parameters such as the retinal output amplitude becomes too large. This is commented below.

## 2.3 Specific features of the model

In addition to propose an original implementation of the retino-cortical V1 system, the model has specific features. First, this is a two dimensional model in space, allowing to play complex trajectories and actually, to simulate the response to realistic visual scenes. This is a feature of the simulator we use, called Macular, developed at Inria. Macular is a simulation platform for the retina and the primary visual cortex (V1), designed to reproduce the response to visual stimuli, in normal vision conditions, or in altered conditions (pharmacology, pathology, development). It is organised into a layered structure that mimics the multi-layers organisation of the retina. It is fed by visual inputs (movies) then processed by the multi-layer structure. A description can be found at https://macular.gitlabpages.inria.fr/macular/user_doc/Macular/main.html.

In contrast to previous models which were considering retino-thalamic entries as spatially periodic functions (rings), the mere fact of introducing realistic retinal inputs leads to a distinction between left and right, bottom and top so that one cannot consider rings geometries anymore. In our case, we need to define proper boundary conditions to the model. Here, we assume zero boundaries conditions for the retinal and the cortical model. In contrast to ring topologies, this induces spatial inhomogeneities, even when considering the rest state.
Finally, why calling this model a "chimera" ? Note that "chimera" is used with different
meanings in the context of biology or computational neuroscience. It can refer to cell
cultures or artificial tissues comprised of cells from multiple individuals (thus having
different DNA) (Madan, 2020). It can also be a dynamical coexistence of coherent and
incoherent states in neuronal network (Majhi et al., 2019). Here, this term is employed
in reference to the Greek mythology where the chimera was a monstrous hybrid creature
composed of various animal parts. Indeed, while the design and parameters of the
cortical model are based on previous works on the monkey visual cortex, the retinal
model is essentially based on previous works dealing with mice retinas. In addition, as
mentioned above, there is the thalamus is considered as a simple relay. In this sense,
our model resembles more a "Frankenstein" creature than a real organism. However, we
believe that it captures important mechanisms in the retino-cortical entanglement. Also,
the advantage of such a model, with its joint simulation platform, is that one can easily
modify the parameters of such or such components and see how it modifies the observed
response. This is actually the main philosophy of this work where we vary physiological
parameters, such as the conduction speed, or the effect of amacrine cells, that cannot
be easily varied experimentally. This also means that the chimera model could be made
closer to monkeys by adjusting the retina model to experimental results on monkeys
retinas. Note that a feature of the simulator Macular is precisely to afford such changes.
<figure><img src="image_3.png" /><figcaption>A retino-cortical model of anticipation</figcaption></figure>
**Figure 1: Synthetic view of the retino-cortical model.** A stimulus is perceived by the retina, triggering a response. **From top to bottom:** The stimulus is first convolved with a spatio-temporal receptive field (black traces labelled "space" and "time" in the purple circles), that mimics, in the Outer Plexiform Layer (OPL,) the concerted activity of Photoreceptors and Horizontal cells, and is fed to Bipolar cells (purple circles). This response is rectified by a low voltage threshold (purple squares). Bipolar cells (BCs) responses are then pooled to Retinal Ganglion cells (RGCs, orange). The firing rate response of a RGC is a sigmoidal function of the voltage (orange square). Gain control can be applied at the BCs and RGCs level (grey circles) triggering anticipation by a shift in the time to peak. The BCs activity is modulated by lateral inhibition through Amacrine cells (ACs, pink). The RGCs response (firing rate) is sent to cortical columns in the primary visual cortex depicted, at the bottom right, as two interconnected mean field units (small circles) corresponding respectively to excitatory (green) and inhibitory (red) population. A sketch of this interconnection is shown on the right zoom (grey ellipses). Cortical columns are connected together by an excitatory (big green circle) and inhibitory (big red circle) horizontal connectivity. Note that we assume the same conduction velocity $v_c$ for both connectivities.

## 2.4 Observables for anticipation

From now, both simulated retinal and cortical areas are 2D spaces of 81 × 15 retinal cells. As two consecutive retinal or cortical columns are spaced by 0.225° of visual angle these areas correspond to 18.225° × 3.375°. Using the conversion factor of about 0.3 mm per degree in the retina, approximated from https://www.ncbi.nlm.nih.gov/books/NBK11556/ and 3 mm per degree in the cortex for humans gives a retinal area of ~ 5.46 × 1 mm² and a cortical area of 54.6 × 10 mm² with a spacing of 67.5 µm between retinal cells and 675 µm between cortical columns. We discard the first and the last horizontal degree in all figures to reduce boundaries effects, giving thus an effective cortical space of 16.45° long.

We mainly consider the motion of a bar, moving horizontally, from left to right, along the *x* axis with a constant speed $v_B$. The bar starts to move at time $t = 0$ where the center of the RF of the cortical column located at $x = 0$ coincides with the middle of the bar. The bar has dimensions 0.67 × 0.9 degrees of visual angle, i.e. its height is small compared to the vertical extent of the retina and the cortex. As a consequence, we will consider, in the definition of anticipation observables, that the bar response is characterized by two dimensional graphs, with one spatial dimension, *x*, and the time, *t*.

### 2.4.1 Retinal anticipation

It is characterized by a temporal shift in the peak of the ganglion cell response to a moving object, occurring before the peak response to the same object when flashed. This can be explained by gain control which has the effect of advancing the peak response of the cell's activity (Berry et al., 1999). In our model, this effect can arise at the level of BCs or RGCs. ACs can also induce an advancement of the peak, independently of gain control, although these two effects can constructively combine. We define two
observables: the time at which the peak of the response occurs (Time To Peak, TTP); and the delay between the arrival of the bar at the centre of the receptive field and the time of the peak (Stationary Peak Delay-SPD). These quantities are defined in detail in the next section, as they are also used to characterize cortical anticipation.

### 2.4.2 Cortical anticipation

Here, we define several observables related to the cortical activity when responding to the moving bar. These observables are introduced in a case where the sensory drive of the cortical model consists of the retinal response to a white bar with neither gain control nor amacrine cells. Thus, the retina is passive. We refer this as control conditions (CTL). The illustrative figure 2 corresponds to a bar moving at 6°/s (equivalent to 18 mm/s in the cortex). This is also the value of the default bar speed.

The indicators for cortical anticipation are based on the typical VSDI signal curves, shown in Fig. 2A, and reproducing the experimental observations made by Benvenuti et al. (2020). We first define quantities attached to individual columns, i.e. depending on the spatial coordinate $x$.

**Local observables.**

* $t_{center}(x)$ is the time when the middle of the bar reaches the center of the receptive field of the cortical column located at $x$. We have thus $t_{center}(0) = 0$.

* **The activation time (AT)**, $t_{ON}(x)$, is the time when the VSDI signal response becomes larger than a threshold $\theta = 0.001$.

* **The latency** is $t_L(x) = t_{ON}(x) - t_{center}(x)$. Since $t_{ON}(x) \le t_{center}(x)$, the latency can be negative or equal to 0. *The latency decreases therefore when its value becomes more negative.*

* **The time to peak (TTP)** is the time $t_P(x)$ when the VSDI signal reaches its maximum (see Fig. 2B).

* **The peak delay** is $t_{PD}(x) = t_P(x) - t_{center}(x)$. This quantity can, a priori, have any sign, although in CTL conditions it is always positive. It can be negative though due to e.g. to the retina influence.

The latency, $t_L(x)$, depends on the distance of the RF of the cortical column to the position where the bar started. To illustrate this dependence we use a color code, similarly to Benvenuti et al. (2020). Actually, Fig. 2B is a representation of what is experimentally observed in their paper. The different RF positions in the visual field are represented by a color gradient (Fig. 2B), from black ($x = 0$ degree, point where the bar starts) to light blue ($x = 18.45$ degree). As shown in Fig. 2B, when plotting all cortical responses for various trajectory lengths, aligned on $t_{center}(x)$ (so that the time variable is $t - t_{center}(x)$), one observes a change in the shape of the response prior to the peak: the latency decreases with the distance to the bar origin, i.e. as the distance of the cortical columns to the bar origin increases, these columns respond earlier and earlier to
the bar propagation. Following (Benvenuti et al., 2020) we interpret this as a cortical
*anticipation by latency*. We now introduce global observables to quantify this form of
anticipation.

**Global spatial observables.** Although the latency corresponds to a time as a function of space, it is useful to invert the axes and, instead, to represent space as a function of time: the curve $t_L(x)$ becomes $x(t_L)$ by a simple symmetry with respect to the first diagonal. The cortical space, plotted in function of the activation time and the time to peak have characteristic shapes illustrated in Fig. 2C. This allows us to define four spatial observable:

* **The anticipation range (AR)** is the maximal spatial limit of anticipation, the maximal distance at which cortical columns interact to anticipate motion. It corresponds to the ordinate of the inflexion point in Fig. 2C. Beyond this spatial position, the working hypothesis is that anticipation by latency saturates, due to the limited cortical extent of horizontal connectivity which causes anticipation (Angelucci et al., 2002). Consequently, beyond the anticipation range, the slope of the activation time becomes constant (Fig. 2C), the activation time is no longer affected by the horizontal cortical propagation, but only by the bar speed.

* **The short-range activation speed (SRAS)** is the slope of the activation time before the inflection point (Fig. 2C). When plotted as a function of time from stimulus onset, this speed actually sums up two different contributions: the bar speed and the anticipation speed carried by horizontal connectivity (Fig. 2C). We subtract the bar speed so that the SRAS corresponds only to the contribution of lateral (intra cortical) connectivity.

* **The Long range activation speed (LRAS)** is the slope of the activation time after the inflection point. This cortical region is characterized by an absence of horizontal cortical connectivity influence. In our model, its slope is only constrained by the bar speed. We mention this quantity because it is expected to vary due to feedback effects, not present in our model (Benvenuti et al., 2020).

* **The Peak speed (PS)** is the slope of the time to peak curve. It is parallel to the moving bar curve.

The figure 2 essentially shows that cortical columns sense the bar earlier than the time
when the bar enters their receptive field. For Benvenuti et al. (2020) this is an indication
that cortical anticipation mainly holds by latency. This is the explanation of the change
in the early VSDI response profile (before the peak), in Fig. 2B, as a function of the
color gradient. One observes that the latency becomes more and more negative with the
distance of the cortical column to the position where the bar starts, (Fig. 2D), meaning
that the cortical columns are informed earlier and earlier that "something" is arriving.
This observation holds until a saturation value, the *minimal latency*, where the gain of
anticipation provided by the horizontal cortical connectivity reaches a maximum. This
suggests that cortical connectivity and axonal velocity impacts anticipation (sections
3.2.3, 3.2.4). As developed below, we also observe an anticipation by peak shift, where
the peak in the simulated VSDI signal is advanced with respect to the peak in the RGCs response.

**Global temporal observables.** One can also plot $x$ as a function of the latency. We obtain two temporal observables, illustrated in Fig. 2D:

* **The minimal latency (ML).** One observes that the latency of cortical columns close from the starting point of the bar (black circles) decreases (becomes more negative), until it saturates, as the columns are located further and further from the starting point of the bar. The asymptotic value is the minimal latency.

* **The Stationary Peak Delay (SPD).** In contrast to ML, the peak delay is actually independent of $x$, in all the cases studied in the paper. We call this value the SPD.
<figure><img src="image_4.png" /><figcaption>Figure 2: **The paradigm of cortical anticipation by latency.** A) **Typical time course of the VSDI signal response** for a fixed cortical column located at x. The time to peak corresponds to the yellow dotted line. The insert shows a zoom of the proximal part of the curve where the activation time is computed (the black dotted line corresponds to the threshold on the activation time, see text.). B) **Global collapse of the VSDI signals.** The curves, corresponding to cortical columns located at different spatial locations, are aligned on $t_{center}(x)$. Then, these curves are colored so that the different receptive fields positions in the visual field are represented by a color gradient (displayed on the top of the figure), from black ($x = 0$ degree, point where the bar starts) to light blue ($x = 18.45$ degree). The legend at the top left represents the color gradient (from black to light blue) associated with the position of the cortical column. C) **Inverted time space representation** where the cortical space (x coordinate) is represented as a function of time. The dotted black line corresponds to the displacement of the center of the moving bar. The *time to peak*, $t_{P}(x)$, is represented by triangles, colored according to the color gradient. The *peak speed* (PS) is the slope of this curve (thus, in CTL conditions, this is the bar speed, $v_{B}$). The activation time, $t_{ON}(x)$ represented by circles colored according to the color gradient exhibits a crossover between two regimes, well fitted by straight lines (red curves), and separated by an inflection point. The spatial position of this inflection point is the *anticipation range* (AR). The slope of the first linear part (dark red</figcaption></figure>
dashed arrow) corresponds to the *short-range activation speed (SRAS)*. The slope of the
second linear part (light red dashed arrow) is the *long range activation speed (LRAS)*. In
CTL conditions, this slope is equal to *v<sub>B</sub>*. **D)** *Minimal latency (ML) and Stationary*
**Peak Delay (SPD)**. The cortical space is represented as a function of the latency (in
ms). The latency decreases until saturation (red vertical line), at a time called *minimal*
*latency (ML)*. The peak delay (triangle) is constant and equal to the Stationary Peak
Delay (SPD), (yellow vertical line).

3 Results

**3.1 Model calibration**

The model was calibrated so as to reproduce the results obtained by Benvenuti et al. (2020) in the monkey primary visual cortex. Note however that, in our simulations, the cortical area and the visual field are bigger than in their experiments (resp. 54 mm versus 21 mm, and 18 degrees versus 7 degrees). A larger simulated cortical area was indeed needed to be able to correctly observe the effect of certain parameters such as bar speed or cortical extension.

The figure 2 illustrates the results of our model in CTL conditions, with a bar speed of 6°/s. It reproduces the shape of the latency curve shown in (Benvenuti et al. (2020), Fig. 4): cortical columns far from the bar origin are activated earlier than the ones close to the bar origin. This curve is divided in two regimes commented in section 2.4 and illustrated in Fig. 2. In the first regime the slope (short-range activation speed) has a value of 24.8°/s in experimental data and of 21.3°/s in simulated data. The inflection point (anticipation range) differs as well: 2° in experiments and 6.3° in simulations. Also, the activation in our model can occur up to 880 ms before the middle of the bar arrives at the center of the receptive field (i.e. a minimal latency of -880 ms) while in (Benvenuti et al., 2020) "latency scatters for the medium and long trajectories that fully covered a wide range of values from 0 to -400 and -600 ms". We finally note a difference in the time of the peak, very close to zero in experiments (Fig. 4D of Benvenuti et al. (2020)) but located at 139 ms in the simulation.

## 3.2 Cortical anticipation in CTL conditions depends on stimulus features and on physiological parameters

**3.2.1 Increasing the retinal output amplitude enhances anticipation**

We varied the retinal output amplitude (ROA), sent by retinal ganglion cells, in a range
from 1 to 50 Hz (figure 3). This range of variations has been chosen from Berry et al.
(1999); Chen et al. (2013); Buffet (2024).

Increasing the ROA has the effect of increasing almost linearly the firing rate of the RGCs
and the simulated VSDI signal (Fig. 3 E) which shows a slight saturation presumably
due to the sigmoid activation functions F<sub>E</sub> , F<sub>I</sub> (eq. (9)). Here, we chose to stay in the
almost linear range. Indeed, we observed that increasing too much the amplitude of the
input leads to pathological oscillations which, as mentioned above, are artefacts of the
cortical model.

The peaks of simulated VSDI signal, shown in Fig. 3 A (1 Hz) and Fig. 3 B (35 Hz), has globally the same shape as the paradigmatic figure 2A, although we observe a clear difference in the latencies between A and B. This is a first indication that increasing the ROA enhances anticipation by latency. Fig. 3 C shows the time course of the simulated VSDI signal for the cortical column located at the center of the layer (x = 9°, y = 1.35°). One observes an increase in the slope before the peak, resulting in a minimal latency (ML) which becomes more negative as shown in Fig. 3 F (red trace). There is a saturation for large ROA though. Along the same lines, we plot, in 3 D, the spatial simulated VSDI signal for the time when the central cortical column reaches its maximum. The *x* coordinates has been shifted so that the central cortical column is located at *x* = 0 in this figure. We observe a spread of the left part of the peak and a more abrupt slope on the right part, as the ROA increases. Those combined effects results in a increase of the anticipation range (AR) as shown in Fig. 3 F (green trace), with again a saturation effect. Finally, the spatial and temporal effects combine to increase the short range anticipation speed (SRAS). Therefore, anticipation by latency becomes more prominent (ML and AR) and spreads faster (SRAS) as the ROA increases.

Interestingly, we also observe a slight anticipation by peak shift in Fig. 3 C. We have plotted, in Fig. 3 H, the SPD as a function of the ROA. It is constant for RGCs, but essentially decreases for the cortex. This figure also shows the difference between the cortical and retinal SPD, which is negative, meaning that the simulated VSDI signal peak arises earlier than the RGC frequency peak. Thus, in addition to show anticipation by latency, the simulated VSDI signal is also a bit in advance on the RGC peak. In Fig. 3I we have plotted the time course of the frequency response around the peak for the central RGC, and, in Fig. 3 J, the time course of the VSDI, around the peak for the central cortical column. The red traces correspond to a 1 Hz ROA and the green traces to 35 Hz. The green traces have actually been rescaled to match the amplitude of the red ones. This is to show that, in addition to a simple rescaling (which makes the RGC response overlap in Fig. 3 I), there are, in the cortex, non linear effects which modify the shape of the response and thereby impact the anticipation.

To summarize, an increase in the ROA non-linearly enhances the anticipation which extends further, earlier and faster, with a saturation when the ROA becomes too large. Rising the ROA increases the overall activity level transmitted horizontally by cortical columns. Thus, more distant cortical columns are above the threshold earlier as shown by the decrease in ML and the increase in AR. The results also demonstrate the presence of anticipation by peak shift increasing with the ROA. However, the peak shift is independent on the distance of the cortical column to the starting point of the bar, in contrast to anticipation by latency.

**Stimulus contrast.** We assessed the impact of the stimulus contrast in the movies by increasing it from 0.1 to 1 by increments of 0.1. As we checked the effect is completely equivalent to increasing the amplitude *C* of the OPL kernel and thus, the ROA (not shown).
<figure><img src="image_5.png" /><figcaption>Figure 3: **The effect of the retinal output amplitude (ROA) on the cortical response.**
A) ROA = 1 Hz; B) ROA = 35 Hz. The color bar on the left of Fig. A represents the color gradient introduced in section 2.4. C) **Temporal simulated VSDI signal** in response to increasing retinal amplitude for the cortical column located at the center of the layer (x = 9°, y = 1.35°). D) **Spatial simulated VSDI signal** in response to increasing retinal amplitude, for the time where the central cortical column reaches its maximum. The x coordinates has been shifted so that the central cortical column is located at x = 0. E) **Simulated VSDI signal amplitude** of the central cortical column versus the ROA. F) **Minimal latency** (red, scale on the right) and anticipation range (green, scale on the left) versus the ROA. G) **Short-range activation speed** (red) versus the ROA. H) **Retino-cortical stationary peak delay (SPD) variation** versus the ROA for the central cell. The ganglion firing rate SPD is plotted in orange, the cortical SPD in blue (scale on the left), and the difference in black (scale on the right). I) **Shape of the central, rescaled, RGC response profile to the moving bar**, for 1 Hz (red) and 35 Hz (dashed green). J) **Simulated VSDI signal**, for 1 Hz (red) and 35 Hz (dashed green). The dotted vertical lines in I, J correspond to the peaks in the RGCs firing rate or simulated VSDI signal at 1 Hz (red) and 35 Hz (green).</figcaption></figure>
3.2.2 Effect of bar speed on cortical anticipation

The level of anticipation certainly depends on the object' speed. To investigate the
respective retinal and cortical impact on this effect we have done simulations with
stimulus speeds ranging from 3 °/s (equivalent to 9 mm/s of the cortex) to 30 °/s
 resp. 90 mm/s in the cortex), first in CTL conditions, with the default parameters
of Appendix 5.1. We actually had to limit the speed range above 3°/s because, in
control conditions, the cortical model is subject to oscillations for lower speed values.
In contrast, we checked that the addition of gain control or sufficiently strong amacrine
cells inhibition eliminates these oscillations and makes it possible to reduce the bar speed
below 3°/s (section 3.3.3). Another issue, though, is the computational time which
becomes prohibitively long when the bar speed is too small.

Our results are summarized in Fig. 4. Increasing the bar speed decreases the retinal output amplitude (ROA). This is because the OPL kernel $K_{B_i}$ in eq. (1) has less time to integrate the stimulus, leading to a decrease in the BCs activity and RGCs activity as the bar speed increases (Fig. 4 A). This induces a decrease in the simulated VSDI signal activity (Fig. 4 B). Thus, one expects a reduction of the anticipation by latency. Increasing the bar speed indeed increases the minimal latency (ML) and decreases the anticipation range (AR) (Fig. 4D).

Is this effect on ML and AR only due to the ROA reduction or are there more subtle, non
linear, effects hidden ? To address this question we plotted ML, AR as a function of the
ROA itself, controlled by the bar speed (light green curve), and compared to the case
where the ROA was under direct control (dark green curve) as in the previous section.
When acting on it via the bar speed, the ROA varies in the interval [10, 45] Hz, a bit less
that the range of ROA explored in the previous section (Fig. 4 E). AR is almost linear
this interval of variation, for both curves. In contrast, ML decreases non linearly, and
goes to a saturation when the ROA is directly modified (red trace) whereas it decreases
sharply and slightly non linearly in the case where the ROA is modulated by the bar
speed (brown trace). This evidences that the decay in ML versus the bar speed is not
only due to the decay in the ROA but includes additional, non linear effects.

Similarly, we studied the effect of the bar speed on the short range activation speed (SRAS), Fig. 4 F. This quantity shows an increase up to 10.5°/s (2.7 mm/s in the retina, 27 mm/s in the cortex), then a decrease, suggesting the existence of a range of preferred speeds where anticipation by latency is optimal. Note that this effect cannot be explained only by the decrease of the ROA, as shown by Fig. 4 G. The red trace (direct control of ROA) is rather different from the curve where ROA is controlled by the bar speed (brown trace).

Finally, we investigated the role of the bar speed on the anticipation by peak shift (Fig. 4 C). The stationary peak delay (SPD) shows up a minimum at about 4.1°/s, where the advance of the cortical SPD with respect to the RGC SPD is maximal. After this minimum there is a non linear, sigmoidal like, increase of the SPD, which switches from negative to positive at about 16°/s. Thus, for larger speed, the cortical peak is delayed with respect to the RGC peak.
<figure><img src="image_6.png" /><figcaption>Figure 4: **The effect of the bar speed on the cortical response in CTL conditions.** A) Decay of the BC voltage peak and RGC firing rate (ROA) as the bar speed increases. B) Simulated VSDI signal amplitude of the central cortical column. C) Stationary peak delay difference between cortical VSDI and RGC firing rate. D) Minimal latency (ML, red) and anticipation range (AR, green) in function of the bar speed. E) ML and AR as a function of the RGC firing rate, in the case where the firing rate is constrained by the ROA (red for ML, dark green for AR) and in the case where the firing rate is constrained by the bar speed (brown for ML, light green for AR). F) Short-range activation speed (SRAS, red) versus bar speed. G) SRAS versus the RGC firing rate, in the case where the firing rate is constrained by the ROA (red) and in the case where the firing rate is constrained by the bar speed (brown).</figcaption></figure>

We observe a strong decrease in retinal activation versus bar speed (Fig. 4 A). This is in contradiction with the behavior measured in the retina in e.g. Fig. 5 a in Berry et al. (1999) or Fig. 5 in Johnston and Lagnado (2015)). In a range of speed from 0.1 to 1 mm/s they observe that peak firing rates of ganglion cells are nearly proportional to speed and thus increase with it. As shown in section 3.3.3 this discrepancy is resolved by adding active retinal anticipation mechanisms.
3.2.3 The role of excitatory and inhibitory connections length on cortical anticipa-
tion

We have next explored the influence of the excitatory and inhibitory connectivity lengths
on cortical anticipation, an effect which cannot be studied experimentally (Fig. 5). We
need to recall first a salient feature of the model. The cortical connectivity is modeled by
a *normalized* Gaussian kernel (section 2.2, eq. (12)) where the cortical extent (excitatory
or inhibitory) is the mean square deviation of the Gaussian. This normalisation was
introduced in the original model (Zerlaut et al., 2018) to ensure that the number of
connections within the system remains constant when the size of the extensions is
increased. As a consequence, the shorter the cortical extensions, the more the Gaussian
connectivity profile is concentrated around the cortical column’s receptive field, with
a higher weight. Inversely, increasing the cortical extensions spreads the Gaussian
and reduces its weights. Therefore, acting on the Gaussian mean square deviation
dramatically influences the value of the mean cortical column voltage for the excitatory
and inhibitory populations, eq. (16), as well as their polarisation in the steady-state (i.e.
without stimulus). These quantities have a significant impact on the simulated VSDI
signal, anticipation range (AR), and minimal latency (ML), as illustrated in Fig. 5.

We maintained the ratio between the two respective lengths to a constant ratio of 0.2
so as to keep the balance between excitation and inhibition extents. This choice relies
on the literature showing that horizontal intra-cortical excitatory axons are longer than
inhibitory axons (Buzás et al., 2006; Kisvárday, 2016; Buzás et al., 2001; Kisvarday
et al., 1997). The excitatory extent was then varied from 1° to 7° and the inhibitory extent
from 0.2° to 1.4°. The first prominent effect, observed in Fig. 5 A, is the behaviour of
the simulated VSDI signal amplitude. It decays up to a minimum at about 2° before
increasing. To better understand this behaviour we have plotted the excitatory and
inhibitory mean-voltages (eq. (16)). In Fig. 5 F, we see the peak of the excitatory mean
voltage µ<sub>V,E</sub> as a function of the excitatory cortical extent (dark green) while the light
green curve represents the excitatory mean voltage in the absence of any stimulus, µ<sub>V,E<sub>0</sub></sub>.
The difference between the two is shown in Fig. 5 H (dark green). Likewise, Fig. 5
G shows the peak of the inhibitory mean voltage (brown) and inhibitory mean voltage
without stimulus (red), while the red trace in Fig. 5 H shows the difference between
these two quantities. The total simulated VSDI signal as a function of the cortical extent
is a linear combination of these two traces, (eq. (21)).

When the cortical extent increases from 1° up to 2° we observe that the excitatory
and inhibitory mean voltages increase while the mean voltage differences decreases,
explaining the observed decay of the simulated VSDI signal. This decrease occurs
because the mean voltage in the steady-state increases faster than the peak mean voltage,
for both populations. As anticipated above, this behaviour can be explained by the
Gaussian connectivity profiles. Increasing the cortical length decreases the intensity
of the coupling between two cortical columns. Short extensions give a concentrated
Gaussian with strong weights but short range influence. When the inhibitory cortical
extent is 0.2° (resp. 1° for the excitatory cortical extent), the inhibitory Gaussian is so
concentrated that its weight far exceeds that of the excitatory. This inhibitory dominance
gives rise to a strong steady-state hyperpolarisation and an overall sensitisation of
the cortical column which reacts more strongly in the presence of a stimulus. This
hyperpolarisation is reduced by the elongation of the inhibitory extensions above 0.2°.
This mean votage difference decreases leading to a reduction in the VSDI. From 2°,
we observe a decay in the steady-state voltage for excitatory and inhibitory population,
corresponding to a new phase of hyperpolarisation in the steady-state. When the cortical
extent increases, the excitatory and inhibitory weights decrease thereby diminishing the
mean voltage at steady-state. In contrast, the mean voltage peak increases for excitators
and have a moderate variation for inhibitors. As a consequence, the mean voltage
differences increase, leading to an increase of the VSDI.
<figure><img src="image_7.png" /><figcaption>Figure 5: **The effect of excitatory and inhibitory connectivity length on the cortical response.** In the whole figure, the variable on abscissa is the excitatory cortical extent, but recall that there is a constant ratio with the inhibitory extent. A) **Simulated VSDI signal amplitude** of the central cortical column versus the cortical extent length. B) **Anticipation range (green) and minimal latency (red)**. The black dotted line represents the equality between distance and length of excitatory connectivity. C) **Short-range activation speed (red)** in function of excitatory extent. D) **Difference between RGC stationary peak delay (SPD) and simulated VSDI signal SPD.** E) **Cortical space** as a function of delay for excitatory extent at 7° and inhibitory extent at 1.4°: latency (circle) and peak delay (triangle). F) **Excitatory mean voltage** at the peak (dark green) and without stimulus (steady-state, light green). G) **Inhibitory mean voltage** at the peak (dark red) and in the steady-state (light red). H) **Mean voltage peak amplitude** for excitators (green) or inhibitors (red). This is the difference between the average peak voltage and the steady-state voltage.</figcaption></figure>
We observe more anticipation by latency since AR increases and ML decreases when the cortical extent increases (Fig. 5 B) as well as short range anticipation speed (SRAS) (Fig. 5 C) with a saturation for an excitatory and inhibitory length respectively at 7° and 1.4°. Note that the measured AR is always larger than the length of the corresponding excitatory connectivity (black dotted line, Fig. 5B). The cortical columns are therefore influenced beyond the excitatory extent, and, therefore also *earlier*, promoting anticipation. This is particularly true up to 4° where this effect is maximal. The stationary peak delay (SPD) is also affected by the increase in cortical extent, Fig. 5 D. For cortical extents smaller than 2° the cortical peak is delayed with respect to the RGC peak. In particular, as the connections length tends to 0, the SPD difference (black trace) in Fig. 5 tends to 0. This shows that the difference between RGC and VSDI SPD is primary due to the cortical horizontal connectivity. Beyond 2° the simulated VSDI signal peak is more and more in advance to the RGC peak with a saturation at about 5°, Fig. 5 D. The time scale of this peak delay (maximum –14 ms) is quite lower than the ML though. This effect is weaker than anticipation by latency and does not evolve along the trajectory of the bar, as shown in Figure 5E. This means that the effect is the same whatever the position of the cortical column on the trajectory of the bar.

This peak shift can be explained by two hypotheses, both resulting from an adapting-like
anticipation mechanism reducing the maximum amplitude of the response and shifting
its peak. The first hypothesis is based on a "cortical" gain control mechanism. Due to
the non linearity of the conductances a change in the baseline activity of the column
can induce a faster drop off of the signal, triggering an advanced peak shift. Note that
the baseline activity depends on the cortical extents. The second hypothesis, which can
cumulate with the previous one, is based on a network effect, and more precisely on
inhibition. When the bar arrives, the inhibitory effects of upstream activated cortical
columns lead to an earlier peak drop off. Both hypotheses actually depend on the cortical
horizontal connectivity and could explain the peak shift decay as the inhibitory cortical
extent decreases.

To conclude, anticipation is enhanced by increasing the length of excitatory and inhibitory fibres at a constant ratio for values inferior to 7° with an earlier shift in the VSDI peak. Unlike anticipation by latency, the peak shift is independent of the trajectory, though. Both effects are highly dependent on changes in the Gaussian profile. They are the strongest with an excitatory length of 4° and an inhibitory length of 0.8°. This is close to the physiological connectivity that we use in the rest of the paper.

**3.2.4 The role of conduction velocity**

We finally investigated the influence of fibre conduction velocity between cortical
columns on anticipation in control conditions (Figure 6), in the range 100 – 333 °/s.
We observed no effect on the simulated VSDI signal (not shown), a slight decrease in
ML, and a slight increase in AR (Fig. 6 A). More interestingly, SRAS is increasing
almost linearly. In their paper, Benvenuti et al. (2020) proposed a phenomenological,
physiologically plausible model of horizontal cortical integration in response to a moving
bar (Fig. 3 of their paper). In their model, SRAS increases linearly with a 1:1 ratio to the
fibre conduction speed (Fig. 3D of their paper). In our model, which integrates more
biological features, we also observe a (quasi)-linear behaviour but the slope, ~0.035 is far from 1. In our opinion, this is because they only used horizontal excitation. However, inhibition acts as in impediment to the activation of cortical columns, explaining our small slope. The increase in fibre conduction speed is also accompanied by a very slight decrease of 0.6 ms in the shift between the cortex and the ganglion cells SPD (6 E).

To sum up, the only remarkable effect induced by an increase in the conduction velocity is a linear increase in the SRAS, similar to what has been conjectured by Benvenuti et al. (2020), but with a quite smaller slope, due to horizontal inhibition.

<figure><img src="image_8.png" /><figcaption>Figure 6: **The effect of fibre conduction velocity on the cortical response**
**A) Anticipation range** (green) and minimal latency (red) in function of fibre conduction velocity.
**B) Short-range activation speed** (red). **C) Difference** between RGC stationary peak delay (SPD) and simulated VSDI signal SPD.</figcaption></figure>

### 3.2.5 Conclusion of section 2.4.2

In normal, biological conditions, the input received by the cortex comes from a retina having active non-linear mechanisms of anticipation. Hence, although our chimera model in CTL conditions can qualitatively reproduce the cortical anticipation observed
experimentally, there are substantial differences. The anticipation in CTL conditions clearly results from the propagation of activity in an excitatory/inhibitory balanced horizontal network. In addition to cortical anticipation latency, the cortical columns exhibit an anticipation mechanism that is independent of latency anticipation and causes an earlier peak offset. This mechanism could be associated with dynamical normalization generated by cortical inhibition and leading to an adapting-like anticipation in the cortex. This is why it is affected by changes in lateral connectivity and ROA. However, such normalization mechanisms are also present in the retina and their role ought to be prominent in tuning anticipation, as we explore now.

**3.3 Cortical anticipation is influenced by the retina**

We now investigate how the cortical response behaves when the retinal drive is itself
generating peak anticipation during motion integration, using two mechanisms: gain
control and inhibition by ACs. Note that, in general, these mechanisms also modify the
time course of the RGCs response to the moving bar (see Fig. 7 I, 8 I). The resulting
effect on the cortical response is thus not only a shift in the simulated VSDI signal peak,
but a change in its global shape as well (see e.g. Fig. 7 J, 8 J or Fig. 9). This entails
potential changes in the simulated VSDI signal amplitude and in the latency.

Gain control and amacrine cells connectivity cause a decrease in the Retinal Output Amplitude (ROA) with an impact on the cortical response. Thus, to isolate the effect of e.g. gain control on anticipation we need to compare the situation with gain control to the situation without it, when *the ROAs are identical*. The set of control simulations where no effect (gain control or amacrine cells network) is present, but where the ROA is rescaled to match the case where the effect is present is called "equivalent retinal output amplitude" (EROA).

**3.3.1 Gain control in the retina enhances cortical anticipation**

In the model, gain control can be present in BCs or RGCs. We ran simulations with increasing the strength of these two gain controls and compare these results to EROA conditions. We only show here the figure for bipolar gain control, which is the most representative; the figure for RGCs gain control can be found in the supplementary figure S1.

**Variation of BCs gain control.** We first simulated the response to the moving bar with a bipolar gain control $h_B \in [0, 9.2]$ mV/s (Fig. 7). We investigate here what is the induced effect on cortical anticipation, by latency and by peak shift.

The first prominent effect of BCs gain control (Fig. 7 B, $h_B = 9.2$ mV/s) is a pronounced shift in the peak in the simulated VSDI signal response followed by a secondary peak. In addition the latencies are slightly smaller than in CTL. The increase in $h_B$ is also associated to a moderate decrease in the amplitude of the simulated VSDI signal (Fig 7 C, D, E). This was expected since the strength of gain control in BCs reduces the amplitude of the retinal response. We observe however that the amplitude of the simulated VSDI signal is slightly higher than in EROA condition, Fig. 7 E. This is because, when $h_B$ increases, gain control changes the shape of the BC response (data not shown) and
thereby the RGC output profile (Fig. 7 I, J). Figures 7 F also shows an increase in ML and a decrease in AR. There is no difference with EROA for AR while the increase in ML is less marked than for EROA condition. This indicates a positive effect of gain control on ML, which partially compensates for the anticipation decay due to retinal amplitude output reduction. SRAS increases much more than in EROA condition (Fig. 7 G). In comparison, it is 26.7% faster than the speed observed in EROA for BCs gain control at 9.2 mV/s. This effect is strong enough to compensate and even reverse the slight decrease due to the reduction in ROA.

Concerning the anticipation by peak shift, the presence of BCs gain control results in a –77 ms (–51.5 %) increase in RGC stationary peak delay (SPD) between $h_B = 0$ and $h_B = 9.2$ mV/s (Fig. 7 I). This is accompanied by an earlier shift in the cortex peak of –72.5 ms (–53 %) (Fig. 7 J). Note the difference in the response profiles for $h_B = 0$ (red traces) and $h_B = 9.2$ mV/s (green dashed traces) in the RGC response and in the VSDI response. Fig. 7 H illustrates the evolution of the RGC SPD (orange trace), the cortical SPD (blue trace), and the difference of these two quantities (black trace).

To sum up, this study demonstrates the direct influence of retinal peak anticipation on the cortex when increasing the BCs gain control: an earlier shift of the RGC peak, a reduced ROA, and a change in the RGC response profile, with a corollary anticipation in the simulated VSDI signal, by peak shift and by latency. However, the impact on anticipation depends on the level of BCs gain control showing that there is an optimal value for BCs gain control. Let us finally remark that the small "rebound" in the simulated VSDI signal observed in Fig. 7 B, after the first peak, could be tracked in experimental VSDI signal. If such a rebound was found, it would not necessarily mean that it comes from BCs gain control, but at least that a mechanism of the same nature holds in the pathway from photoreceptors to V1. This could actually also help calibrating better chimera like models.

**Variation of RGCs gain control.** We also varied the RGCs gain control, $h_G \in [0, 0.54]$ mV$^{-1}$ Hz (Supplementary figure S1). Note that the range of values is very different from BCs gain control because the non linearity in the gain function is quite different (see eq. (5) versus (8)). The result shows similarities with BCs gain control though. RGCs gain control induces as well an earlier shift of the RGC peak, a reduced ROA, and a change in the RGCs response profile, with a corollary anticipation in the simulated VSDI signal, by peak shift and by latency. We have however less cortical anticipation through latency : ML increases and AR decreases while SRAS increases until a maximum. ML and SRAS vary less than in EROA, showing that the RGCs gain control generates a compensation of the ROA decay. Finally, in contrast to BCs gain control (Fig. 7 B) there is no rebound in the simulated VSDI signal after the first peak (see the comment at the end of the previous section).
<figure><img src="image_9.png" /><figcaption>Figure 7: **The effect of bipolar gain control on the cortical response.** Simulated VSDI signal response for A) $h_B = 0$ mV/s (control) and B) $h_B = 9.2$ mV/s. C) Simulated VSDI signal in response to increasing bipolar gain control for the cortical column located at the center of the layer. D) Spatial simulated VSDI signal in response to increasing bipolar gain control, for the time where the central cortical column reaches its maximum. In C and D color correspond to different values of $h_B$ (in mV per seconds, labelled "mVps"). E) Simulated VSDI signal amplitude of the central cortical column versus $h_B$. F) Anticipation range (green) and minimal latency (red) versus $h_B$. G) Short-range activation speed (red) versus $h_B$. In A, B, C, we also drawn the Equivalent Retinal Output Amplitude (EROA) curve. This is the dotted black curve with the same coloured symbols. H) Stationary peak delay for RGCs (orange), simulated VSDI signal (blue) for the central cell (scales on the left) and difference between RGC SPD and simulated VSDI signal SPD (black, scales on the right). I) Shape of the central RGC response profile to the moving bar, without gain control (red) and with BC gain control $h_B = 9.165$ mV/s (dashed green). Note that the two traces have been rescaled to have the same maximum. J) Simulated VSDI signal, same conditions. In I,J, the dotted lines correspond to the peaks in the RGC firing rate or simulated VSDI signal without BCs gain control (red) and with it (green).</figcaption></figure>
3.3.2 The anticipatory role of amacrine cells

ACs, like horizontal cells, provide a lateral connectivity entangling the "vertical" infor-
mation channels from photoreceptors to retinal ganglion cells via bipolar cells. They
constitute networks which modulate and can potentially propagate the response of BCs
to a moving signal to other BCs and RGCs. This depends though crucially on the scaling
between synaptic weights ($w_B^A, w_A^B, w_G^A, w_G^B$), and characteristic integration times ($\tau_A, \tau_B, \tau_G$). The network response to the moving bar also depends on the connectivity
structure. Here, we analyse the impact on cortical anticipation, of feedforward and
feedback inhibitions, although we didn't make a systematic study of these effects when
varying the control parameters. As a matter of fact, our conclusions here are limited
to one set of parameters values providing examples of what could happen in the early
visual system when ACs lateral connectivity takes place.

**Amacrine feedforward inhibition.** We study first the direct influence of feedforward inhibition pathway with connectivity from BCs to ACs and from ACs to RGCs, respectively characterized by the parameters $w_A^B$ and $w_G^A$ in eq. (2). We varied $w_G^A \le 0$, controlling the intensity of ACs synapses to RGCs, and $w_A^B \ge 0$, controlling the intensity of BCs synapses to ACs, from 0 to 1 Hz, restricting to the case $w_G^A = -w_A^B$. The other parameters are tuned to the control value (appendix 5.1). The results are shown in Fig. 8.
First, ACs influence the VSDI response as shown in Fig. 8 B for $|w_G^A| = w_A^B = 1$ Hz with an important increase in latency, a narrower main peak than in CTL (Fig. 8 A), followed by a small rebound. As expected, the inhibitory effect of ACs induces a decrease in the simulated VSDI signal amplitude (Fig. 8 C, D, E) but also a decrease in AR (Fig. 8 D) and an increase in ML (Fig. 8 C). The simulated VSDI signal curve in this condition is essentially indistinguishable from the EROA curve (Fig. 8 E), confirming that the effect of ACs on VSDI, in feedforward conditions, is only a drop in the ROA. However, Fig. 8 F shows also a detrimental effect of feedforward inhibition on AR, as well as on the ML, more important than in EROA. In contrast, SRAS shows a compensatory effect of the retinal amplitude (Fig. 8 G). While SRAS at EROA decreases brutally, the SRAS with feedforward amacrine connectivity remains relatively constant suggesting a mechanism which counterbalances the decrease in speed due to the decrease in amplitude.

The most prominent effect occurs on the SPD of RGCs (orange trace) and simulated
VSDI signal (blue trace) which decrease sharply (Fig. 8 H), though with a difference
becoming smaller and smaller in absolute value (black trace). This is accompanied by
a strong advancement of -104 ms for the RGC SPD (Fig. 8 I) and of -88 ms for the
simulated VSDI signal peak (Fig. 8 J). We also observe a small rebound of the simulated
VSDI signal (Fig. 8 J), arising when $|w_G^A|$ becomes large. When the bar arrives in their
receptive field, BCs activity increases, increasing the ACs activity and their inhibitory
effect on the RGC. When $|w_G^A|$ is large enough it takes over the excitation from BCs
and the RGC firing rate drops to 0. Because the ACs have here a shorter time scale than
BCs (corresponding e.g. to the peak of the green dashed curve in Fig. 8 I), their effect
lasts shorter and the rebound corresponds to the residual activity coming from BCs.

Thus, the presence of feedforward inhibitory connectivity enhances the anticipation by
peak shift within the cortex, with an increasing effect as the intensity of synaptic weights
increases, while penalising AR and ML. However, the speed of the anticipatory wave
remains stable thanks to compensation for the decay of the ROA. All these effects are
not simply due to a decrease in the ROA but entail additional effects presumably due, as
for gain control, to the difference in the BCs response profile.

<figure><img src="image_10.png" /></figure>

Figure 8: **The effect of feedforward inhibition,** controlled by $w_G^A = -w_G^B$, on the cortical response. **Simulated VSDI signal response** with: A) $|w_G^A| = 0$ Hz (control) and B) $|w_G^A| = 1$ Hz. C) **Temporal simulated VSDI signal** in response to increasing $|w_G^A|$ for the central cortical column. D) **Spatial simulated VSDI signal** for the time where the central cortical column reaches its maximum. E) **Simulated VSDI signal amplitude** of the central cortical column. F) AR (green) and ML (red). G) SRAS (red). H) SPD for RGCs (orange), cortical columns (blue) and difference between the cortical and RGC SPD (black). **J) Shape of the central RGC response profile to the moving bar,** when $w_G^A = -w_A^B = 0$) (red) and when $|w_G^A| = w_A^B = 1$ Hz, (dashed green). Note that the two traces have been normalized to have the same maximum at 1 Hz. **I) Simulated VSDI signal,** same conditions. The dotted lines in E, F correspond to the peaks in the RGC firing rate or simulated VSDI signal without feedforward AC connection (red) and with it (green).
**Amacrine feedback inhibition.** The feedback loop BCs → ACs → BCs induce a complex interaction between the moving object, the local cells response and the influence of these cells on distant cells, propagated via the lateral amacrine inhibition. The picture is that of a moving bar propagating in a non homogeneous landscape of activities modulated by the amacrine network. This entails specific effects, not present with a feedforward connectivity, such as the existence of resonant frequencies (Cessac, 2022), a change in the shape of BCs response (Kartsaki et al., 2024) or the existence of a preferred speed range at which anticipation by adaptation is maximal (Ebert, 2023). In addition, the feedback loop propagates inhibition producing a decay of the response before and after the peak. This actually differs from gain control, which only acts on the post-peak portion of the curve. These feedback effects are controlled by the product $w_B^A w_A^B$ and by the characteristic times $\tau_A, \tau_B$. Here, we keep $\tau_A, \tau_B$ constant and vary $w_A^B$ with $w_B^A = -w_A^B$ from 0 to 12 Hz (Supplementary figure S2).

In short, the results demonstrate the capacity of amacrine retro-control to provoke an increasing anticipation by peak shift influencing the cortex. However, this is accompanied by a negative effect on cortical anticipation by latency mainly manifested by a rise in ML and a drop in the AR and of the cortical peak shift. In contrast, the anticipatory wave speed is increasing a little bit. In general, these effects were much less pronounced than for feedforward inhibition, and no second peak was observed in the time course of the VSDI. These conclusions are not general though as a more exhaustive study would require to vary other parameters as well, a task well beyond the scope of this paper.

### 3.3.3 Active normalisation in the retina provides the correct behaviour of RGCs firing rates versus bar speed

At the end of section 3.2.2 we pointed out the wrong behaviour of the retinal activation versus bar speed, in CTL conditions. In contrast, we expect to have the correct behaviour in the presence of gain control. Indeed, our results show first an increase of the activation rate of RGCs, for a limited speed range [0.1 – 1] mm/s ([0.33 – 3.3] °/s) (Berry et al., 1999; Johnston and Lagnado, 2015), and then a decrease. This is shown in Fig. S3 B,C in agreement with what is observed experimentally by Summers and Feller (2022). A similar behaviour is also obtained with amacrine cells (Fig. S3 D, E).

In addition, we check that gain control or a sufficiently strong amacrine cells inhibition eliminates the pathological oscillations reported in section 3.2.2 and makes it possible to study the effects of bar speeds below 3°/s, although simulation time becomes prohibitively long.

### 3.3.4 The anticipatory impact of the retina

We now synthesize our observations about the cortical correlates of the retinal anticipatory mechanisms. Although our investigations are absolutely non exhaustive - a more detailed study would require to vary a larger set of parameters in the model - it reveals several effects which are generic, i.e. present on a wide range of parameters value, although the *quantitative* observations may depend on these parameters. These generic effects are an advancement of the RGC output peak, a decrease in the ROA, and a global
change in the shape of the firing rate response. This has an impact on the VSDI response, as illustrated in the figures above and in Fig. 9.

In Fig. 9 A, we compare the respective effect of BCs and RGCs gain control on the simulated VSDI signal at the central cortical column. As a systematic study would require to modify jointly $h_B$ and $h_G$ in a two dimensional map, a task beyond the scope of this paper, we only compare the model response in a case where BCs and RGCs gain control are tuned so that the retinal output amplitude is the same (ROA = 17.1 Hz), that is $h_B = 8.554$ mV/s (magenta trace) and $h_G = 0.359$ mV/s (yellow trace). The first observation is a peak shift stronger for BC gain control than for RGC gain control. There is a SPD difference of 25 ms. There is also a strong difference in the simulated VSDI signal profile, after the pic, with a secondary peak in the case of BCs gain control. It is unfortunately not possible to compare this with the experimental results in (Benvenuti et al., 2020) as the VSDI profiles they show (e.g. Fig. 4 D) is cut before the place where such a possible rebound may occur. Performing new experiments on a larger time scale would be a way to confirm the role of BCs gain control on the experimental VSDI signal profile. BCs gain control appears thus more effective at generating peak anticipation while limiting the impact on cortical latency anticipation and increasing the speed of anticipation.

Fig. 9 B compares the role of feedforward versus feedback amacrine cells connectivity, adjusting the control parameters $w_G^A = 0.7$ Hz (feedforward) and $w_B^A = 11$ Hz (feedback) so that the ROA are equal to 13.1 Hz. We observe here a small pic shift and a small change in latency while the main visible effect is a change in the shape of the VSDI profile after the peak. A comparison of Fig. 8 and Fig. S2 suggests a better efficiency of amacrine connectivity in the form of a feedback loop. Anticipation by peak shift is more pronounced while limiting the impact on cortical latency anticipation despite the smaller intrinsic cortical delay.
<figure><img src="image_11.png" /><figcaption>Figure 9: **Gain control and amacrine connectivity comparison.** A) **Time course** of the VSDI response with a BC gain control $h_B = 8.554$ mV/s (magenta) and a RGC gain control $h_G = 0.359$ mV/s (yellow). B) **Time course** of the VSDI response with amacrine feedback, $|w_B^A| = 11$ Hz (purple) and feedforward, $|w_G^A| = 0.7$ Hz (orange). C) **Time course** of the VSDI response with bipolar gain control $h_B = 8.554$ mV/s (magenta) and amacrine feedback connectivity weight $|w_B^A| = 11$ Hz (purple). D) **Time course** of the VSDI response with ganglion gain control $h_G = 0.359$ mV/s (yellow) and amacrine feedforward connectivity weight $|w_G^A| = 0.7$ Hz (orange).</figcaption></figure>

Fig. 9 C compares the BCs gain control to the ACs feedback still tuning the respective parameters so that the ROA are equal ($h_B = 9.2$ mV/s, $|w_B^A| = 9$ Hz, same ROA of 16.1 Hz). In this example BCs gain control provides more anticipation by peak with a visible effect on latency. Fig. 9 D compares as well the RGCs gain control to the ACs feedforward connectivity with $h_G = 0.36$ mV/s, $w_A^B = 0.6$ Hz (ROA = 17 Hz). For this set of parameters RGC gain control performs better than feedforward amacrine connectivity.

A synthesis of our results is presented in table 2, in the appendix, summarizing the effects of active retinal mechanisms on each anticipation observable and comparing them to experimental and control conditions. Yet, the main difficulty of drawing conclusions
about the "best" circuit configurations is that we studied the anticipation mechanisms independently, while they are certainly coupled in the real brain. For example, the condition showing the best match with experiments is with feedforward amacrine - ganglion cells with a weight of 0.8 Hz. The minimum latency is -523 ms (-400 to -600 ms in experiments), the anticipation range is 3.74 ° (2 ° in experiments), the short-range activation speed is 24 °/s (24.8 °/s in experiments). Nevertheless, the delay of the peak of the cortical response remains at 80 ms instead of the 0 ms in experiments, and increasing the weight of amacrine cells towards the ganglion cells brings us further away from biology. To compensate this, it would be better to set up simultaneously amacrine inhibition and gain control on both bipolar cells and ganglion cells. These two effects could however either cooperate or interfere with each other, depending on parameters. The addition of these effects would make it possible to reduce the cortical response time to 0 ms. But, then, a corollary detrimental effect would be a reduction of the peak amplitude. In the real brain, all these effects have been tuned together by evolution and adaptive mechanisms such as homeostasy and plasticity that are clearly missing in our model.

# 4 Discussion

In this paper, we presented a "chimera" construction connecting a multi-layered model of the retina to a mean-field model of V1. This allowed us to investigate how the retino-cortical pathway (where the thalamus-LGN was considered as a simple relay) can anticipate the trajectory of a moving object, while offering the ability to separately tune retinal and cortical effects.

Cortical anticipation is first manifested by a spatial gradient in the latency of the simulated VSDI signal. This corresponds to a set of cooperative and cumulative non linear mechanisms where the cortical columns sensing the bar send information feedforward to pre-activate distant columns, generating what we called "anticipation by latency". However, as we have shown, this mechanism is not sufficient *per se*. To match experimental observations, it is necessary to incorporate retina mechanisms (gain control and amacrine connectivity) where the retina actively cooperates with the cortex V1 to anticipate the trajectory. Essentially, this leads to a strong advancement of the retinal ganglion cell response peak, with a deformation of the whole retino-cortical signal. Although the mechanism of peak anticipation is known since long in the retina (Berry et al., 1999), our work is the first, to our knowledge, showing how the effect of the peak offset in the retina is transmitted to the cortical response. This leads to a first question: We can understand how the peak shift contributes to anticipation, but what about the role of graded latency slopes? Do they have a specific role in anticipation compared to the peak shift ? Could it be possible that the peak shift informs that "something has happened" while anticipation by latency would inform the next columns that "something will be arriving" ? We could also imagine the existence of a population of neurons within visual areas beyond V1 that would be sensitive to the slope rather than the peak. In that case, these neurons would be activated well before the peak occurs and would therefore make it possible to strongly anticipate the position of the moving object.
Throughout this article, we have also been interested in determining to which degree specific individual mechanisms contribute to motion anticipation, by varying their intensity. Parameters such as the amplitude of the retinal output, the excitatory and inhibitory cortical connectivity extent, gain control intensity or amacrine cells connectivity strongly impact the anticipation, although it is difficult to determine if one mechanism dominates. A conclusion could certainly be drawn for *our set of parameters*. For example, we observe that bipolar gain control and feedforward amacrine connectivity are more efficient than ganglion cells gain control or feedback amacrine connectivity, but we cannot draw general conclusions simply because the potential space of parameters is very large and our simulations correspond to a small region in this space. In addition, we tuned these parameters separately, while, in the biological realm, they have been tuned together by evolution and are presumably adjusted by homeostasis and plasticity. They may also change with species, age, and so on. Presumably, different anticipation mechanisms are there because they contribute together, in a way that remains to be understood. From this perspective, our study is a first step toward deeper investigations.

There is actually an additional known mechanism that we haven't considered in the main text: the so-called "predicting anticipation" discovered recently by Menz et al. (2020). They observed a rise in the RGCs activity before a moving object enters their receptive field, advancing the entire retinal response (not only the peak), with a strong anticipatory effect. The authors hypothetised that this mechanism is due to the lateral inhibition by biphasic ACs. These cells have a first negative part of their activity that depolarizes BCs, followed by a second positive part that inhibits them. These two parts are relatively symmetrical and very close in amplitude. We managed to reproduce this mechanism by modifying our model to include two populations of amacrine cells (Supplementary figure S2). Although this result is preliminary, it shows that sending such a signal to V1 would result in an increase in the early part of the peak and a similar reduction in the late part. This would result in a shift of the VSDI response arising earlier with a constant amplitude at the peak since we have equivalent stimulation and inhibition.

Another message of this work is that different retinal mechanisms have very similar effects, in a way which should be difficult to disentangle from experimental cortical recordings. This is apparent in Fig. 9. A notable exception is the presence of a "rebound" in the simulated VSDI signal which signals retinal anticipation. As we mentioned in the paper, it could be interesting to track such rebounds in experimental recordings. Another avenue of exploration could be to investigate in more detail the influence of the object speed on the response. The existence of a range of preferred speed for anticipation could provide interesting insights about the structure and tuning of the underlying networks. Also, as we argued, a moving object triggers, in the retina and in the cortex, waves of activity which should influence the shape of the response, providing as well indications on the underlying networks.

We would like now to discuss some possible extensions of the model. A first extension would be to consider center surround receptive fields for bipolar cells. A moving bar extended in the direction perpendicular to motion will significantly stimulate the surround of bipolar cells, leading to effects that we didn't explore in this paper. Another important improvement would be to integrate a realistic thalamus, that is, adding thalamic mean-
field equations so as to mimic the retino-thalamo (LGN)-cortical pathway (Overwiening et al., 2024). Most excitatory synapses in the thalamus are of cortical origin (Sherman and Guillery, 2011), and thus the cortico-thalamic feedback, a feature not present in our model, is essential. In addition, thalamic neurons have very complex intrinsic firing properties and can generate bursts of action potentials so they are also not a simple relay. A future step would be to integrate these properties of the LGN by using a previously proposed model of the thalamus using AdEx neurons, which are formulated in mean-field (Destexhe, 2009) and integrated in our simulator. This would lead us to simulate the role of the thalamo-cortical loop not only for anticipation. Also, the cortical model exhibit pathological instabilities, when the retinal output amplitude increases too much. We have shown that these instabilities are reduced by adding active retinal control mechanisms. This suggests that existing model of the visual system should take these mechanisms in consideration. These instabilities can also be controlled either by considering a second order mean-field model (ElBoustani and Destexhe, 2009; Aquilué-Llorens et al., 2024) or an adaptive mean-field model (Di Volo et al., 2019).

The retina model could be extended through an implementation of other cell types and more diverse types of connectivity, more realistic than nearest neighbours (Souihel and Cessac, 2019). This would allow to implement more realistic retinal circuits, and assess their role in motion processing and anticipation. Comparing the respective role of the parvocellular and magnocellular pathways could also be insightful. Magnocellular cells have a very wide receptive field and a very short characteristic response time. They are sensitive to movement and blind to colours. The magnocellular pathway carries therefore information about large, fast moving objects. Parvocellular cells have a narrow receptive field and a large integration time. They are sensitive to shapes and colours. The parvocellular pathway carries therefore information about small, slow objects (Zeki, 2015). How do these two pathways, with a different sensitivity to motion, contribute to anticipation ?

This opens up questions about how the heterogeneous structure of the retina, with multiple cells sub-types and a large palette of synapses, contributes to motion anticipation and more generally to motion processing. In particular, the retina has the ability to detect surprise (Schwartz et al., 2007; Ebert et al., 2024; Despotović et al., 2024), e.g. an abrupt change in a trajectory. How is the surprise information transmitted to the visual cortex, how is it used, and how could it be measured? These are all interesting questions that could be explored in future studies.

**Acknowledgments.** This work was supported by the National Research Agency (ANR), in the project "Trajectory", https://anr.fr/Project-ANR-15-CE37-0011, funding Selma Souihel's PhD; the ANR too in the project "Shooting Star-15755" https://anr.fr/Projet-ANR-20-CE37-0018, funding Jérôme Emonet's PhD, and finally by the interdisciplinary Institute for Modelling in Neuroscience and Cognition (NeuroMod http://univ-cotedazur.fr/en/idex/projet-structurant/neuromod) of the Université Côte d'Azur. Mattéo Di Volo and Alain Destexhe are supported by the CNRS and the European Union (The Human Brain Project, H2020-945539). We thank O. Marre
A retino-cortical model of anticipation

and S. Chemla for helpful discussions. We are grateful to the Service d'Etudes et
Développement at Inria for helping develop the software Macular.
# 5 Appendix

## 5.1 Model parameters (default values)

<table><thead><tr><th>Parameter category</th><th>Parameter Name</th><th>Symbol</th><th>Value</th><th>Unit</th></tr></thead><tbody><tr><td rowspan="2">Simulated area</td><td>Number of cells (X)</td><td>n<sub>cellsX</sub></td><td>83</td><td>Dimensionless</td></tr><tr><td>Number of cells (Y)</td><td>n<sub>cellsY</sub></td><td>15</td><td>Dimensionless</td></tr><tr><td></td><td>Width (X)</td><td>L<sub>X</sub></td><td>18.45</td><td>degrees</td></tr><tr><td></td><td>Height (Y)</td><td>L<sub>Y</sub></td><td>3.15</td><td>degrees</td></tr><tr><td></td><td>Cell distance</td><td>δ</td><td>0.225</td><td>degrees</td></tr><tr><td></td><td>mm retina per degree</td><td>rpd</td><td>0.3</td><td>mm/degrees</td></tr><tr><td></td><td>mm cortex per degree</td><td>cpd</td><td>3</td><td>mm/degrees</td></tr><tr><td>Stimulus video (moving bar)</td><td>Speed</td><td>v<sub>B</sub></td><td>6</td><td>degrees/s</td></tr><tr><td></td><td>Width (X)</td><td>l<sub>X</sub></td><td>0.67</td><td>degrees</td></tr><tr><td></td><td>Height (Y)</td><td>l<sub>Y</sub></td><td>0.9</td><td>degrees</td></tr><tr><td></td><td>Frame rates</td><td>δ<sub>t</sub></td><td>60</td><td>Hz</td></tr><tr><td></td><td>Pixel per degree</td><td>ppd</td><td>300</td><td>pixel/degrees</td></tr><tr><td rowspan="2">OPL</td><td>OPL input amplitude</td><td>C</td><td>0.025</td><td>Dimensionless</td></tr><tr><td>Integration time step (ODE solver)</td><td>dt</td><td>0.0004</td><td>s</td></tr><tr><td></td><td>RF size</td><td>σ<sub>c</sub></td><td>0.2</td><td>degrees</td></tr><tr><td></td><td>Characteristic time</td><td>τ<sub>c</sub></td><td>0.1</td><td>s</td></tr><tr><td rowspan="2">Bipolar cells</td><td>Characteristic time</td><td>τ<sub>B</sub></td><td>0.1</td><td>s</td></tr><tr><td>Activity characteristic time</td><td>τ<sub>aB</sub></td><td>0.1</td><td>s</td></tr><tr><td></td><td>Threshold</td><td>θ<sub>B</sub></td><td>0</td><td>mV</td></tr><tr><td></td><td>Activity rate</td><td>h<sub>B</sub></td><td>[0,9.2]</td><td>mV<sup>-1</sup>Hz</td></tr><tr><td rowspan="2">Amacrine cells</td><td>Characteristic time</td><td>τ<sub>A</sub></td><td>0.05</td><td>s</td></tr><tr><td>Weight bipolar to amacrine</td><td>|w<sup>B</sup><sub>A</sub>|</td><td>[0,12]</td><td>Hz</td></tr><tr><td></td><td>Weight amacrine to bipolar</td><td>w<sup>A</sup><sub>B</sub></td><td>[0,12]</td><td>Hz</td></tr><tr><td></td><td>Weight amacrine to ganglion</td><td>w<sup>A</sup><sub>G</sub></td><td>[0,1]</td><td>Hz</td></tr><tr><td rowspan="2">Ganglion cells</td><td>Radius bipolar to ganglion</td><td>σ<sub>G</sub></td><td>0.3</td><td>degrees</td></tr><tr><td>Characteristic time</td><td>τ<sub>G</sub></td><td>0.1</td><td>s</td></tr><tr><td></td><td>Activity characteristic time</td><td>τ<sub>aG</sub></td><td>0.189</td><td>s</td></tr><tr><td></td><td>Threshold</td><td>θ<sub>G</sub></td><td>0</td><td>mV</td></tr><tr><td></td><td>Activity rate</td><td>h<sub>G</sub></td><td>[0,0.54]</td><td>mV<sup>-1</sup>Hz</td></tr><tr><td></td><td>Weight bipolar to ganglion</td><td>w<sup>B</sup><sub>G</sub></td><td>0.15</td><td>Dimensionless</td></tr></tbody></table>
A retino-cortical model of anticipation

<table><tr><td>Pooling extent</td><td>σ</td><td>0.09</td><td>mm</td></tr><tr><td>Non linear rectification</td><td>α<sub>G</sub></td><td>1110</td><td>Hz/mV</td></tr><tr><td>Non linear rectification limit</td><td>N<sub>Gmax</sub></td><td>212</td><td>Hz</td></tr><tr><td rowspan="10">Cortical shared parameters values</td><td>Membrane capacitance for cortical columns</td><td>C<sub>m</sub></td><td>0.2</td><td>nF</td></tr><tr><td>Excitatory reversal potential</td><td>V<sub>E</sub></td><td>0</td><td>mV</td></tr><tr><td>Inhibitory reversal potential</td><td>V<sub>I</sub></td><td>-80</td><td>mV</td></tr><tr><td>Leak reversal potential</td><td>V<sub>L</sub></td><td>-65</td><td>mV</td></tr><tr><td>Cell number in network</td><td>N<sub>tot</sub></td><td>10 000</td><td>Dimensionless</td></tr><tr><td>Excitatory quantal conductance</td><td>Q<sub>E</sub></td><td>1.5</td><td>nS</td></tr><tr><td>Leak conductance</td><td>g<sub>L</sub></td><td>10</td><td>nS</td></tr><tr><td>Excitatory characteristic time</td><td>τ<sub>E</sub></td><td>0.005</td><td>s</td></tr><tr><td>Inhibitory characteristic time</td><td>τ<sub>I</sub></td><td>0.005</td><td>s</td></tr><tr><td>External drive</td><td>ν<sub>ext</sub></td><td>2</td><td>Hz</td></tr><tr><td>Mean field characteristic time</td><td>T</td><td>0.005</td><td>s</td></tr><tr><td>Fraction of inhibitory cells</td><td>g<sub>ei</sub></td><td>0.2</td><td>Dimensionless</td></tr><tr><td>Probability of connectivity</td><td>p<sub>connec</sub></td><td>0.0375</td><td>Dimensionless</td></tr><tr><td>Mean voltage initial fit value</td><td>μ<sub>V</sub><sup>0</sup></td><td>-60</td><td>mV</td></tr><tr><td>Mean voltage initial fit variation</td><td>δμ<sub>V</sub><sup>0</sup></td><td>10</td><td>mV</td></tr><tr><td>Sigma initial fit value</td><td>σ<sub>V</sub><sup>0</sup></td><td>4</td><td>mV</td></tr><tr><td>Sigma initial fit variation</td><td>δσ<sub>V</sub><sup>0</sup></td><td>6</td><td>mV</td></tr><tr><td>Normalized tau initial fit value</td><td>(τ<sub>V</sub><sup>N</sup>)<sup>0</sup></td><td>0.5</td><td>Dimensionless</td></tr><tr><td>Normalized tau initial fit variation</td><td>(δτ<sub>V</sub><sup>N</sup>)<sup>0</sup></td><td>1</td><td>Dimensionless</td></tr><tr><td rowspan="3">Cortical excitators (RS)</td><td>Sigma extent</td><td>σ<sub>E</sub></td><td>1.67</td><td>degrees</td></tr><tr><td>Initial activity</td><td>ν<sub>E0</sub></td><td>1.86</td><td>Hz</td></tr><tr><td>Inhibitory quantal conductance</td><td>Q<sub>I</sub></td><td>3</td><td>nS</td></tr></table>
<table><caption>Table 1: **Model parameters.** For each parameter in our retino-cortical model, we give its name, symbol, unit and the default value used in control condition.</caption><tbody><tr><td></td><td>P parameters</td><td>P<sub>E</sub></td><td>[-49.8, 5.06, -25, 1.4, -0.41, 10.5, -36, 7.4, 1.2, -40.7]</td><td>mV</td></tr><tr><td rowspan="3">Cortical inhibitors (FS)</td><td>Sigma extent</td><td>σ<sub>I</sub></td><td>0.3</td><td>degrees</td></tr><tr><td>Initial activity</td><td>ν<sub>I0</sub></td><td>12.66</td><td>Hz</td></tr><tr><td>Inhibitory quantal conductance</td><td>Q<sub>I</sub></td><td>5</td><td>nS</td></tr><tr><td></td><td>P parameters</td><td>P<sub>I</sub></td><td>[-51.4, 4, -8.3, 0.2, -0.5, 1.4, -14.6, 4.5, 2.8, -15.3]</td><td>mV</td></tr><tr><td rowspan="7">Cortical column connectivity</td><td>Retino-cortical amplitude</td><td>w<sub>RC</sub></td><td>2.5</td><td>Dimensionless</td></tr><tr><td>Cortical density</td><td>ρ<sub>cort</sub></td><td>4000</td><td>mm<sup>-2</sup></td></tr><tr><td>Retinal density</td><td>ρ<sub>ret</sub></td><td>400</td><td>mm<sup>-2</sup></td></tr><tr><td>Fiber conduction velocity</td><td>v<sub>C</sub></td><td>300</td><td>mm/s</td></tr><tr><td>Weight excitatory to excitatory</td><td>A<sub>E</sub><sup>E</sup></td><td>1</td><td>Dimensionless</td></tr><tr><td>Weight inhibitory to inhibitory</td><td>A<sub>I</sub><sup>I</sup></td><td>1</td><td>Dimensionless</td></tr><tr><td>Weight excitatory to inhibitory</td><td>A<sub>I</sub><sup>E</sup></td><td>1.5</td><td>Dimensionless</td></tr><tr><td></td><td>Weight inhibitory to excitatory</td><td>A<sub>E</sub><sup>I</sup></td><td>1</td><td>Dimensionless</td></tr></tbody></table>
## 5.2 Synthesis of anticipation mechanisms effects on observables

<table><caption>Table 2: **Synthesis of anticipation mechanisms effects.** First column: anticipation observables. Second column Values measured in experiments. CTL Simulated values in control conditions. BC GC: variations of observables when $h_B$ increases. RGC GC: variations of observables when $h_G$ increases. AC FF: variations of observables when the AC feedforward parameter increases. AC FB: variations of observables when the AC feedback parameter increases.</caption><thead><tr><th>Observable</th><th>Exp.</th><th>CTL</th><th>BC GC</th><th>RGC GC</th><th>AC FF</th><th>AC FB</th></tr></thead><tbody><tr><td>AR (°)</td><td>2</td><td>6.3</td><td>6.3 → 5.7</td><td>6.2 → 5.5</td><td>5.9 → 2.7</td><td>6.6 → 4.8</td></tr><tr><td>ML (ms)</td><td>[-600,-400]</td><td>-880</td><td>-880 → -803</td><td>-868 → -767</td><td>-858 → -121</td><td>-895 → -649</td></tr><tr><td>SRAS (°)</td><td>24.8</td><td>21.2</td><td>21.2 → 25.1</td><td>21.2 → 22.5</td><td>22.3 → 26.8</td><td>21 → 23.5</td></tr><tr><td>SPD (ms)</td><td>0</td><td>137</td><td>137 → 64</td><td>129 → 85</td><td>136 → 49</td><td>136 → 87</td></tr></tbody></table>

## References

Angelucci, A., Levitt, J., Walton, E., Hupe, J., Bullier, J., and Lund, J. (2002). Circuits for local and global signal integration in primary visual cortex. *The Journal of Neuroscience*, 22(19):8633–8646.

Aquilué-Llorens, D., Goldman, J. S., and Destexhe, A. (2024). High-Density Exploration of Activity States in a Multi-Area Brain Model. *Neuroinformatics*, 22(1):75–87.

Baccus, S. A., Ölveczky, B. P., Manu, M., and Meister, M. (2008). A retinal circuit that computes object motion. *Journal of Neuroscience*, 28(27):6807–6817.

Benvenuti, G., Chemla, S., Boonman, A., Masson, G., and Chavane, F. (2015). Anticipation of an approaching bar by neuronal populations in awake monkey v1. *Journal of Vision*.

Benvenuti, G., Chemla, S., Boonman, A., Perrinet, L., Masson, G. S., and Chavane, F. (2020). Anticipatory responses along motion trajectories in awake monkey area v1. *bioRxiv*.

Berry, M., Brivanlou, I., Jordan, T., and Meister, M. (1999). Anticipation of moving stimuli by the retina. *Nature*, 398(6725):334–338.

Buffet, T. (2024). *Mechanisms of motion processing in the retina*. phdthesis, Sorbonne Université.

Buzás, P., Eysel, U. T., Adorján, P., and Kisvárday, Z. F. (2001). Axonal topography of cortical basket cells in relation to orientation, direction, and ocular dominance maps. *The Journal of Comparative Neurology*, 437(3):259–285.
Buzás, P., Kovács, K., Ferecskó, A. S., Budd, J. M. L., Eysel, U. T., and Kisvárday, Z. F. (2006). Model-based analysis of excitatory lateral connections in the visual cortex. *The Journal of Comparative Neurology*, 499(6):861-881.

Carlu, M., Chehab, O., Dalla Porta, L., Depannemaecker, D., Héricé, C., Jedynak, M., Köksal Ersöz, E., Muratore, P., Souihel, S., Capone, C., Zerlaut, Y., Destexhe, A., and Di Volo, M. (2020). A mean-field approach to the dynamics of networks of complex neurons, from nonlinear Integrate-and-Fire to Hodgkin-Huxley models. *Journal of Neurophysiology*, 123(3):1042-1051.

Cessac, B. (2022). Retinal processing: Insights from mathematical modelling. *Journal of Imaging*, 8(1).

Chemla, S. (2010). *A biophysical cortical column model for optical signal analysis*. PhD thesis, School of Information and Communication Sciences.

Chemla, S., Chavane, F., Vieville, T., and Kornprobst, P. (2007). Biophysical cortical column model for optical signal analysis. In Holmes, W. R., Jung, R., and Skinner, F., editors, *Sixteenth Annual Computational Neuroscience Meeting (CNS)*, volume 8, Suppl 2 of *BMC Neuroscience*.

Chemla, S., Reynaud, A., di Volo, M., Zerlaut, Y., Perrinet, L., Destexhe, A., and Chavane, F. (2019). Suppressive traveling waves shape representations of illusory motion in primary visual cortex of awake primate. *Journal of Neuroscience*, 39(22):4282-4298.

Chen, E. Y., Marre, O., Fisher, C., Schwartz, G., Levy, J., da Silviera, R. A., and Berry, M. (2013). Alert response to motion onset in the retina. *Journal of Neuroscience*, 33(1):120-132.

Despotović, D., Joffrois, C., Marre, O., and M.Chalk (2024). Encoding surprise by retinal ganglion cells. *PLOS Computational Biology*, 20(4):1-20.

Destexhe, A. (2009). Self-sustained asynchronous irregular states and up-down states in thalamic, cortical and thalamocortical networks of nonlinear integrate-and-fire neurons. *J Comput Neurosci.*, 27(3):493-506.

Di Volo, M., Romagnoni, A., Capone, C., and Destexhe, A. (2019). Biologically realistic mean-field models of conductance-based networks of spiking neurons with adaptation. *Neural Computation*, 31(4):653-680.

DiVolo, M. and Férézou, I. M. (2021). Nonlinear collision between propagating waves in mouse somatosensory cortex. *Scientific Reports*, 11:19630.

DiVolo, M., Romagnoni, A., Capone, C., and Destexhe, A. (2019). Biologically Realistic Mean-Field Models of Conductance-Based Networks of Spiking Neurons with Adaptation. *Neural Computation*, 31(4):653-680.

Ebert, S. (2023). *Dynamical synapses in the retinal network*. PhD thesis, Université Côte d'Azur.
Ebert, S., Buffet, T., Sermet, B., Marre, O., and Cessac, B. (2024). Temporal pattern
recognition in retinal ganglion cells is mediated by dynamical inhibitory synapses.
*Nature Communication*, 15:6118.

ElBoustani, S. and Destexhe, A. (2009). A master equation formalism for macroscopic modeling of asynchronous irregular activity states. *Neural computation*, 21(1):46–100.

Guo, K., Robertson, R., Pulgarin, M., Nevado, A., Panzeri, S., Thiele, A., and Young, P. (2007). Spatio-temporal prediction and inference by v1 neurons. *European Journal of Neuroscience*, 26:1045–1054.

Jancke, D., Erlaghen, W., Schöner, G., and Dinse, H. (2004). Shorter latencies for motion trajectories than for flashes in population responses of primary visual cortex. *Journal of Physiology*, 556:971–982.

Johnston, J. and Lagnado, L. (2015). General features of the retinal connectome deter-
mine the computation of motion anticipation. *Elife*.

Kartsaki, E., Hilgen, G., Sernagor, E., and Cessac, B. (2024). How does the inner retinal network shape the ganglion cells receptive field : a computational study. *Neural Computation*, 36(6):1041–1083.

Kisvárday, Z. (2016). *Topography of excitatory cortico-cortical connections in three main tiers of the visual cortex : functional implications of the patchy horizontal network*, chapter 7, pages 135–158. Elsevier Science.

Kisvárday, Z., Toth, E., Rausch, M., and Eysel, U. (1997). Orientation-specific relation-
ship between populations of excitatory and inhibitory lateral connections in the visual
 cortex of the cat. *Cerebral Cortex*, 7(7):605.

Madan, K. (2020). Natural human chimeras: A review. *European Journal of Medical Genetics*, 63(9):103971.

Majhi, S., Bera, B. K., Ghosh, D., and Perc, M. (2019). Chimera states in neuronal networks: A review. *Physics of Life Reviews*, 28:100–121.

Menz, M. D., Lee, D., and Baccus, S. A. (2020). Representations of the amacrine cell population underlying retinal motion anticipation. *bioRxiv*.

Overwiening, J., Tesler, F., Guarino, D., and Destexhe, A. (2024). A multi-scale study of thalamic state-dependent responsiveness. *bioRxiv*.

Papoulis, A. (1965). *Probability, Random Variables and Stochastic Processes*. McGraw-
Hill, New York.

Schwartz, G., Harris, R., Shrom, D., and Berry, M. J. (2007). Detection and prediction of periodic patterns by the retina. *Nature Neuroscience*, 10(5):552–554.

Sherman, S. and Guillery, R. (2011). Distinct functions for direct and transthalamic corticocortical connections. *J Neurophysiol.*, 106(3):1068–77.
Souihel, S. and Cessac, B. (2019). Anticipation in the retina and the primary visual cortex: towards an integrated retino-cortical model for motion processing. In *ICMNS 2019 - The 5th International Conference on Mathematical NeuroScience*, Cophenhague, Denmark.

Souihel, S. and Cessac, B. (2021). On the potential role of lateral connectivity in retinal anticipation. *J. Math. Neurosc.*, 11(3).

Strauss, S., Korympidou, M. M., Ran, Y., Franke, K., Schubert, T., Baden, T., Berens, P., Euler, T., and Vlasits, A. L. (2022). Center-surround interactions underlie bipolar cell motion sensitivity in the mouse retina. *Nature Communications*, 13(1):5574. Publisher: Nature Publishing Group.

Subramaniyan, M., Ecker, A. S., Patel, S. S., Cotton, R. J., Bethge, M., Pitkow, X., Berens, P., and Tolias, A. S. (2018). Faster processing of moving compared with flashed bars in awake macaque v1 provides a neural correlate of the flash lag illusion. *Journal of Neurophysiology*.

Summers, M. T. and Feller, M. B. (2022). Distinct inhibitory pathways control velocity and directional tuning in the mouse retina. *Current biology: CB*, 32(10):2130-2143.e3.

Veltz, R. (2011). An analytical method for computing Hopf bifurcation curves in neural field networks with space-dependent delays. *Comptes Rendus Mathematique*, 349:749-752.

Zeki, S. (2015). Area v5—a microcosm of the visual brain. *Frontiers in Integrative Neuroscience*, 9.

Zerlaut, Y., Chemla, S., Chavane, F., and Destexhe, A. (2018). Modeling mesoscopic cortical dynamics using a mean-field model of conductance-based networks of adaptive exponential integrate-and-fire neurons. *Journal of Computational Neuroscience*.
# 6 Supplementary Material

<figure><img src="image_12.png" /><figcaption>Figure S1: **Effect of RGCs gain control on the cortical response.** Simulated VSDI signal A) with $h_G$ = 0 mV/s or B) with $h_G$ = 0.54 mV/s. C) **Temporal simulated VSDI signal** in response to increasing gain control for the central cortical column. D) **Spatial simulated VSDI signal** in response to increasing $h_G$ for the time where the central cortical column reaches its maximum. E) **Simulated VSDI signal amplitude** of the central cortical column versus $h_G$. F) **AR (green) and ML (red)** versus $h_G$. G) **SRAS (red)** versus RGC $h_G$. In A, B, C, the dotted black curve with the same coloured symbols corresponds to EROA curve. H) **SPD** for RGCs (orange), simulated VSDI signal (blue) for the central cell (scales on the left) and difference between RGC SPD and simulated VSDI signal SPD (black, scales on the right). I) **Shape of the central RGC response to the moving bar**, without gain control (red) and with $h_G$ = 0.54 mV/s (dashed green). Note that the two traces have been rescaled to have the same maximum. J) **Simulated VSDI signal**, same conditions. In E,F, the dotted lines correspond to the peaks in the RGC firing rate or simulated VSDI signal without RGCs gain control (red) and with it (green).</figcaption></figure>
<figure><img src="image_13.png" /><figcaption>Figure S2: The effect of ACs-BCs feedback loop on the cortical response. Simulated **VSDI signal response** A) without amacrine feedback inhibition (control) or, B) with $|w_B^A| = w_A^B = 12$ Hz. C) **Temporal simulated VSDI signal** in response to increasing amacrine feedback inhibition for the central cortical column. D) **Spatial simulated VSDI signal** in response to increasing amacrine feedback inhibition for the time where the central cortical column reaches its maximum. E) **Simulated VSDI signal amplitude** of the central cortical column versus the feedback loop weight. F) AR (green) and ML (red). G) SRAS (red) in function of amacrine to bipolar weight. H) SPD for RGCs (orange), cortical columns (blue) and difference between the cortical and RGC SPD (black). I) **Shape of the central RGC response profile to the moving bar**, without feedback AC connection ($w_B^A == 0$) (red) and when $|w_B^A| = 12$ Hz, (dashed green). The two traces have been rescaled to have the same maximum. J) **simulated VSDI signal**, same conditions. The dotted lines in E, F correspond to the peaks in the RGC firing rate or simulated VSDI signal without feedforward AC connection (red) and with it (green).</figcaption></figure>
<figure><img src="image_14.png" /><figcaption>Figure S3: **The role of active anticipation mechanisms** on bipolar response and RGC firing rates. **A)** Control condition. **B)** Gain control on BCs with $h_G = 9.165$ mV/s. **C)** Gain control on RGCs with $h_G = 0.538$ mV/s. **D)** Amacrine inhibition on BCs with a weight of 10Hz. **E)** Amacrine inhibition on RGCs with a weight of 0.8Hz.</figcaption></figure>
<figure><img src="image_15.png" /><figcaption>Figure S4: **Predicting anticipation in the chimera model.** A) **Modification of the circuits** to include biphasic ACs, noted $A_{B_i}$, whose profile is formed by a feedforward input from BCs and a feedforward inhibition from another population of ACs, noted A. **B-E) Time profile** of RGCs firing rate (B) and voltage (C), $A_{B_i}$-amacrine voltage (D), and A-amacrine voltage (E) when the biphasic amacrine circuit is inactive (control, black) or activate ("Predictive", red). The grey rectangles represent the time during which the bar is in the receptive field of the cortical column. The black dotted line is the time when the center of the bar passes through the center of the receptive field. **(F, G)** **Simulated VSDI signals** in control condition (F) and when the ACs circuit is activated (G). The red line corresponds to the peak time of the simulated VSDI signal in predicting condition.</figcaption></figure>