<figure><img src="image_1.png" /></figure>

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# The retina as a dynamical system

Bruno Cessac

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Bruno Cessac. The retina as a dynamical system. Dynamics Days 2021 Nice, Aug 2021, Nice, France. hal-03331814

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Submitted on 2 Sep 2021

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# The retina as a dynamical system

B. Cessac, Université Côte d'Azur, INRIA,
Biovision team and Neuromod Institute
Sophia Antipolis, France
# The retina and its dynamics
# The visual system

<figure><img src="image_2.png" /></figure>

## Retina's layered structure
# The visual system

## « Encoder »

### Retina's layered structure

<figure><img src="image_3.png" /></figure>

## « Decoder »

## « Information » flux

<figure><img src="image_4.png" /></figure>
# The visual system

## « Encoder »

### Retina's layered structure

<figure><img src="image_5.png" /></figure>

## « Decoder »

## « Information » flux

<figure><img src="image_6.png" /></figure>
# The visual system

## « Encoder »

<figure><img src="image_7.png" /></figure>

## « Decoder »

<figure><img src="image_8.png" /></figure>

## « Information » flux
# The visual system

## Retina's layered structure

<figure><img src="image_9.png" /></figure>

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

Most neurons in the retina do not spike (except Gcells) =>

« Analogic » computing.
# The visual system

## Retina's layered structure

<figure><img src="image_11.png" /></figure>

Stimuli are non-stationary, with spatio-temporal correlations.
A modeller point of view

1. The retina is a **high dimensional, non autonomous and noisy** dynamical system, layered and structured, with **non stationary and spatially inhomogeneous entries** (visual scenes).

2. **Local circuits** of non linear cells, connected with non linear synapses efficiently process the **local visual information**. These local circuits are connected together, spanning the whole retina in a **regular tiling**.

3. The conjunction of non linearities and regular structure can produce a highly complex **spatio-temporal** dynamics.

4. The retina is sensitive to **changes in its physiology** (development, pharmacology, pathologies). These changes can be addressed from the point of view of **bifurcations theory**.
<figure><img src="image_12.png" /></figure>
# Retinal anticipation

## Random connectivity and motion anticipation

- S. Souihel (PhD 2016-2019), Biovision team,
  INRIA, Sophia Antipolis

- F. Chavane, INT, Marseille

- A. Destexhe, UNIC, Paris

- O. Marre, Institut de la Vision, Paris

- B. Cessac, Biovision team, INRIA, Sophia Antipolis

AGENCE NATIONALE DE LA RECHERCHE
ANR

Project : ANR Trajectory.
Visual Anticipation

**Anticipation** is carried out by the primary visual cortex (V1) through an activation wave

<figure><img src="image_13.png" /></figure>

Source : Benvenutti et al. 2015
Visual Anticipation

**Anticipation** also takes place in the retina

<figure><img src="image_14.png" /></figure>

<figure><img src="image_15.png" /></figure>

Source :
Berry et al.
1999

<figure><img src="image_16.png" /></figure>

<figure><img src="image_17.png" /></figure>

<figure><img src="image_18.png" /></figure>

<figure><img src="image_19.png" /></figure>
Building a 2D retina model for motion anticipation

Souihel S., Cessac B., "On the potential role of lateral connectivity in retinal anticipation", Journal of Mathematical Neuroscience, BioMed Central, 2021, 11, [arXiv:10.1186/s13408-020-00101-z](https://arxiv.org/abs/10.1186/s13408-020-00101-z)

<figure><img src="image_20.png" /></figure>

<figure><img src="image_21.png" /></figure>
Building a 2D retina model for motion anticipation

Souihel S., Cessac B., "On the potential role of lateral connectivity in retinal anticipation", Journal of Mathematical Neuroscience, BioMed Central, 2021, 11, [arXiv:10.1186/s13408-020-00101-z](https://arxiv.org/abs/10.1186/s13408-020-00101-z)

<figure><img src="image_22.png" /></figure>

<figure><img src="image_23.png" /></figure>
# Building a 2D retina model for motion anticipation

Souihel S., Cessac B., "On the potential role of lateral connectivity in retinal anticipation", Journal of Mathematical Neuroscience, BioMed Central, 2021, 11, https://doi.org/10.1186/s13408-020-00101-z

$$ 
\left\{ 
\begin{aligned}
\frac{dV_{B_i}}{dt} &= -\frac{1}{\tau_B}V_{B_i} + \sum_{j=1}^{N_A} W_{B_i}^{A_j} V_{A_j} + F_{B_i}(t), \\
\frac{dV_{A_j}}{dt} &= -\frac{1}{\tau_A}V_{A_j} + \sum_{i=1}^{N_B} W_{A_j}^{B_i} R_{B_i}, \\
\frac{dA_{B_i}}{dt} &= -\frac{A_{B_i}}{\tau_a} + h_B \mathcal{N}(V_{B_i}).
\end{aligned}
\right. 
$$

<figure><img src="image_24.png" /></figure>
# Building a 2D retina model for motion anticipation

Souihel S., Cessac B., "On the potential role of lateral connectivity in retinal anticipation", Journal of Mathematical Neuroscience, BioMed Central, 2021, 11, https://doi.org/10.1186/s13408-020-00101-z

Stimulus dependent drive

$$ 
\begin{cases}
\frac{dV_{B_i}}{dt} = -\frac{1}{\tau_B}V_{B_i} + \sum_{j=1}^{N_A} W_{B_i}^{A_j} V_{A_j} + F_{B_i}(t), \\
\frac{dV_{A_j}}{dt} = -\frac{1}{\tau_A}V_{A_j} + \sum_{i=1}^{N_B} W_{A_j}^{B_i} R_{B_i}, \\
\frac{dA_{B_i}}{dt} = -\frac{A_{B_i}}{\tau_a} + h_B \mathcal{N}(V_{B_i}).
\end{cases}
$$

<figure><img src="image_25.png" /></figure>
# Building a 2D retina model for motion anticipation

Souihel S., Cessac B., "On the potential role of lateral connectivity in retinal anticipation", Journal of Mathematical Neuroscience, BioMed Central, 2021, 11, https://doi.org/10.1186/s13408-020-00101-z

$$ 
\begin{cases}
\frac{dV_{B_i}}{dt} = -\frac{1}{\tau_B}V_{B_i} + \sum_{j=1}^{N_A} W_{B_i}^{A_j} V_{A_j} + F_{B_i}(t), \\
\frac{dV_{A_j}}{dt} = -\frac{1}{\tau_A}V_{A_j} + \sum_{i=1}^{N_B} W_{A_j}^{B_i} R_{B_i}, \\
\frac{dA_{B_i}}{dt} = -\frac{A_{B_i}}{\tau_a} + h_B \mathcal{N}(V_{B_i}).
\end{cases}
$$

Non linear gain control

<figure><img src="image_26.png" /></figure>
# Building a 2D retina model for motion anticipation

Souihel S., Cessac B., "On the potential role of lateral connectivity in retinal anticipation", Journal of Mathematical Neuroscience, BioMed Central, 2021, 11, 10.1186/s13408-020-00101-z

$$ \frac{dV_{B_i}}{dt} = -\frac{1}{\tau_B}V_{B_i} + \sum_{j=1}^{N_A} W_{B_i}^{A_j} V_{A_j} + F_{B_i}(t), $$

$$ \frac{dV_{A_j}}{dt} = -\frac{1}{\tau_A}V_{A_j} + \sum_{i=1}^{N_B} W_{A_j}^{B_i} R_{B_i}, $$

$$ \frac{dA_{B_i}}{dt} = -\frac{A_{B_i}}{\tau_a} + h_B \mathcal{N}(V_{B_i}). \quad \text{Non linear gain control} $$

<figure><img src="image_27.png" /></figure>

<figure><img src="image_28.png" /></figure>

<figure><img src="image_29.png" /></figure>

<figure><img src="image_30.png" /></figure>

<figure><img src="image_31.png" /></figure>

<figure><img src="image_32.png" /></figure>

<figure><img src="image_33.png" /></figure>
# Building a 2D retina model for motion anticipation

Souihel S., Cessac B., "On the potential role of lateral connectivity in retinal anticipation", Journal of Mathematical Neuroscience, BioMed Central, 2021, 11, https://doi.org/10.1186/s13408-020-00101-z

## Inhibition

$$ 
\begin{cases}
\frac{dV_{B_i}}{dt} = -\frac{1}{\tau_B}V_{B_i} + \sum_{j=1}^{N_A} W_{B_i}^{A_j} V_{A_j} + F_{B_i}(t), \\
\\
\frac{dV_{A_j}}{dt} = -\frac{1}{\tau_A}V_{A_j} + \sum_{i=1}^{N_B} W_{A_j}^{B_i} R_{B_i}, \\
\\
\frac{dA_{B_i}}{dt} = -\frac{A_{B_i}}{\tau_a} + h_B \mathcal{N}(V_{B_i}).
\end{cases}
$$

<figure><img src="image_34.png" /></figure>
Building a 2D retina model for motion anticipation

Souihel S., Cessac B., "On the potential role of lateral connectivity in retinal anticipation", Journal of Mathematical Neuroscience, BioMed Central, 2021, 11, https://doi.org/10.1186/s13408-020-00101-z

$$
\left\{
\begin{aligned}
\frac{dV_{B_i}}{dt} &= -\frac{1}{\tau_B}V_{B_i} + \sum_{j=1}^{N_A} W_{B_i}^{A_j} V_{A_j} + F_{B_i}(t), \\
\frac{dV_{A_j}}{dt} &= -\frac{1}{\tau_A}V_{A_j} + \sum_{i=1}^{B} W_{A_j}^{B_i} R_{A_j}, \\
\frac{dA_{B_i}}{dt} &= -\frac{A_{B_i}}{\tau_a} + h_B \mathcal{N}(V_{B_i}).
\end{aligned}
\right.
\quad \text{Excitation}
$$

<figure><img src="image_35.png" /></figure>
# Building a 2D retina model for motion anticipation

Souihel S., Cessac B., "On the potential role of lateral connectivity in retinal anticipation", Journal of Mathematical Neuroscience, BioMed Central, 2021, 11, 10.1186/s13408-020-00101-z

Linear regime

$$ \frac{d\vec{X}}{dt} = \mathcal{L}\vec{X} + \vec{F}(t). $$

$$ \mathcal{L} = \begin{pmatrix} -\frac{I_{N,N}}{\tau_B} & W_B^A & 0_{N,N} \\ W_A^B & -\frac{I_{N,N}}{\tau_A} & 0_{N,N} \\ h_B I_{N,N} & 0_{N,N} & -\frac{I_{N,N}}{\tau_a} \end{pmatrix} $$

<figure><img src="image_36.png" /></figure>
# Building a 2D retina model for motion anticipation

Souihel S., Cessac B., "On the potential role of lateral connectivity in retinal anticipation", Journal of Mathematical Neuroscience, BioMed Central, 2021, 11, 10.1186/s13408-020-00101-z

Linear regime

$$ \frac{d\vec{\chi}}{dt} = \mathcal{L}\vec{\chi} + \vec{F}(t). $$

$$ \mathcal{L} = \begin{pmatrix} -\frac{I_{N,N}}{\tau_B} & W_B^A & 0_{N,N} \\ W_A^B & -\frac{I_{N,N}}{\tau_A} & 0_{N,N} \\ h_B I_{N,N} & 0_{N,N} & -\frac{I_{N,N}}{\tau_a} \end{pmatrix} $$

<figure><img src="image_37.png" /></figure>
Building a 2D retina model for motion anticipation

Souihel S., Cessac B., "On the potential role of lateral connectivity in retinal anticipation", Journal of Mathematical Neuroscience, BioMed Central, 2021, 11, [arXiv:10.1186/s13408-020-00101-z](https://arxiv.org/abs/10.1186/s13408-020-00101-z)

<figure><img src="image_38.png" /><figcaption>Fig. 7. Starburst amacrine cells as simulated with lucifer yellow in wholemount rabbit retina.</figcaption></figure>

<figure><img src="image_39.png" /></figure>

Dynamics is controlled by the spectrum of L which, itself, is controlled by the connectivity matrices.

When these matrices commute the eigenvalues/eigenvectors of L can be expressed via the common eigenvalues/eigenvectors of the connectivity matrices.
Building a 2D retina model for motion anticipation

Souihel S., Cessac B., "On the potential role of lateral connectivity in retinal anticipation", Journal of Mathematical Neuroscience, BioMed Central, 2021, 11, https://doi.org/10.1186/s13408-020-00101-z

<figure><img src="image_40.png" /></figure>

<figure><img src="image_41.png" /></figure>
Building a 2D retina model for motion anticipation

Souihel S., Cessac B., "On the potential role of lateral connectivity in retinal anticipation", Journal of Mathematical Neuroscience, BioMed Central, 2021, 11, https://10.1186/s13408-020-00101-z

<figure><img src="image_42.png" /></figure>

With nearest neighbours interactions eigenvalues are given in terms of the Laplace operator/matrix.

For a certain range of coupling there is a **wave of activation** triggered by a moving stimulus and **ahead of** it.

<figure><img src="image_43.png" /></figure>
Building a 2D retina model for motion anticipation

Souihel S., Cessac B., "On the potential role of lateral connectivity in retinal anticipation", Journal of Mathematical Neuroscience, BioMed Central, 2021, 11, [10.1186/s13408-020-00101-z](https://doi.org/10.1186/s13408-020-00101-z)

<figure><img src="image_44.png" /><figcaption>**Figure 15:** Effect of anticipatory mechanisms on a parabolic trajectory. A) Response to a dot moving along a parabolic trajectory. The first line shows the stimulus, the second line shows the GCells response with gain control, the third line presents the effect of lateral ACells connectivity with $w = 0.3 \text{ ms}^{-1}$, and the last line shows the effect of asymmetric gap junctions with $v_{gap} = 9 \text{ mm/s}$. B) Time course response of a cell responding to the dot near the trajectory turning point. Linear response corresponds to the response to the stimulus without gain control.</figcaption></figure>
# Building a 2D retina model for motion anticipation

Souihel S., Cessac B., "On the potential role of lateral connectivity in retinal anticipation", Journal of Mathematical Neuroscience, BioMed Central, 2021, 11, https://doi.org/10.1186/s13408-020-00101-z

<figure><img src="image_45.png" /></figure>

New form of retinal anticipation induced by the network.

Strong potential implications (e.g. detecting surprise in a visual scene)
Building a 2D retina model for motion anticipation

Souihel S., Cessac B., "On the potential role of lateral connectivity in retinal anticipation", Journal of Mathematical Neuroscience, BioMed Central, 2021, 11, [arXiv:10.1186/s13408-020-00101-z](https://arxiv.org/abs/10.1186/s13408-020-00101-z)

<figure><img src="image_46.png" /><figcaption>Fig. 7. Starburst amacrine cells are simulated with lucifer yellow in wholemount rabbit retina.</figcaption></figure>

<figure><img src="image_47.png" /></figure>
Building a 2D retina model for motion anticipation

Souihel S., Cessac B., "On the potential role of lateral connectivity in retinal anticipation", Journal of Mathematical Neuroscience, BioMed Central, 2021, 11, [arXiv:10.1186/s13408-020-00101-z](https://arxiv.org/abs/10.1186/s13408-020-00101-z)

<figure><img src="image_48.png" /></figure>

Eigenvalues and eigenvectors, in this case, are complex.

<figure><img src="image_49.png" /></figure>

Eigenmodes contributions can interfere in
constructive or destructive way.

<figure><img src="image_50.png" /></figure>

Waves of anticipation can still be observed
in more narrow range of parameters than
the Laplacian case.

<figure><img src="image_51.png" /></figure>

A large part of the spectral and waves
properties come from the fact that weights
are independent.
Mathematical extensions

* Non linear effects (gain control, rectification) maintain the dynamics in a compact set even when some eigenvalues of L are unstable ;

* This non linear dynamical system can be approximated by a piecewise linear one where dynamics is linear in convex domains spanning the phase space;

* The evolution under the influence of a time-dependent stimulus can be computed as a composition of (non commuting) linear operators labelled by their domains of continuity ;
Perspectives

* A retino-cortical (V1) model of anticipation (with F. Chavane, S. Chemla, A. Destexhe, S. Souihel, M. Di Volo).
* A retino-cortical (V1) model of anticipation (with F. Chavane, S. Chemla, A. Destexhe, S. Souihel, M. Di Volo.

<figure><img src="image_52.png" /><figcaption>Figure 11: Comparing the spatio-temporal activity with the different drives. Left column figures display the spatio-temporal retinal activity with different anticipatory mechanisms, and the right column displays the corresponding spatio-temporal cortical activity. The red dashed lines denote the start and the end times of the LN retinal and cortical activities, as a reference base.</figcaption></figure>
# Logiciel Macular

<figure><img src="image_53.png" /><figcaption>## Modèle rétino-cortical de Macular</figcaption></figure>

- Neurones corticaux inhibiteurs

- Neurones corticaux excitateurs

- Cellules ganglionnaires

- Cellules amacrine

- Cellules bipolaires

<figure><img src="image_54.png" /><figcaption>## Simulation du modèle de Macular</figcaption></figure>
Perspectives

* A retino-cortical (V1) model of anticipation (with F. Chavane, S. Chemla, A. Destexhe, S. Souihel, M. Di Volo).

* Is our model able to anticipate more complex trajectories, with accelerations for instance ?

* How to better define connectivity using biology ?

* Would it be possible to design psycho-physical tests clearly showing the role of the retina in visual anticipation ?

* How do a moving object + lateral connectivity shape the spatio-temporal correlations in ganglion cells spike trains emitted toward the brain ?
Linear response for spiking neuronal networks with unbounded
memory

R. Cofré, CIMFAV, Facultad de Ingenieria,
Universidad de Valparaiso, Valparaiso, Chile

Ignacio Ampuero, Departamento de Informatica,
Universidad Tecnica Federico Santa Maria,
Valparaiso, Chile
<figure><img src="image_55.png" /></figure>
Spontaneous spiking activity

<figure><img src="image_56.png" /></figure>
Spontaneous spiking activity

<figure><img src="image_57.png" /></figure>
<figure><img src="image_58.png" /></figure>
<figure><img src="image_59.png" /><figcaption>Sample averaging (non stationary)</figcaption></figure>
<figure><img src="image_60.png" /></figure>

<figure><img src="image_61.png" /></figure>

Variation in the correlation $C_{12,15}(t, t+2)$

Correlation variations are triggered by the stimulus, but they are constrained by the network dynamics.

<figure><img src="image_62.png" /></figure>
How to compute the spatio-temporal response of a neuronal
network to a time dependent stimulus ?

Can we compute the time variation of spike-observables as a function of
the stimulus and spontaneous dynamics ?

B. Cessac, I. Ampuero, R. Cofre, Linear response for spiking neuronal networks with unbounded memory. Entropy, MDPI, 2021, 23 (2), pp.155.
Sub-threshold dynamics:

$$C_k \frac{dV_k}{dt} + g_k(t, \omega) V_k = i_k(t, \omega).$$

* Linear in V.

* Spike history-dependent.

## Dynamics integration

$$\Gamma_k(t_1, t, \omega) = e^{-\frac{1}{C_k} \int_{t_1}^{t} \tau_k(t, \omega) g_k(u, \omega) du}$$

$$V_k(t, \omega) = \underbrace{V_k^{(sp)}(t, \omega) + V_k^{(S)}(t, \omega) + V_k^{(noise)}(t, \omega)}_{V_k^{(det)}(t, \omega)}$$

$$V_k^{(sp)}(t, \omega) = V_k^{(syn)}(t, \omega) + V_k^{(L)}(t, \omega),$$

$$V_k^{(syn)}(t, \omega) = \frac{1}{C_k} \sum_{j=1}^{N} W_{kj} \int_{\tau_k(t, \omega)}^{t} \Gamma_k(t_1, t, \omega) \alpha_{kj}(t_1, \omega) dt_1$$
Variable length Markov chain with unbounded memory

$$P_n[\omega(n) | \omega_{-\infty}^{n-1}] \equiv \Pi \left( \omega(n), \frac{\theta - V_k^{(\text{det})}(n-1, \omega)}{\sigma_k(n-1, \omega)} \right)$$

<figure><img src="image_63.png" /></figure>
B. Cessac, I. Ampuero, R. Cofre, Linear response for spiking neuronal networks with unbounded memory. Entropy, MDPI, 2021, 23 (2), pp.155.

GIF model without stimulus ⇒ *stationary Gibbs distribution*.

*time-dependent stimulus S(t) ⇒ non stationary Gibbs distribution.*

*How is the average of an observable f(ω, t) affected by the stimulus ?*

If S is weak enough: $\delta\mu[f(t)] = [\kappa_f * S](t)$, (linear response).

$$ \kappa_{k,f}(t-t_1) = \frac{1}{C_k} \sum_{r=-\infty}^{n=[t]} C^{(sp)} \left[ f(t, \cdot), \frac{\mathcal{H}_k^{(1)}(r, \cdot)}{\sigma_k(r-1, \cdot)} \int_{\tau_k(r-1, \cdot)}^{r-1} S_k(t_1) \Gamma_k(t_1, r-1, \cdot) dt_1 \right] $$
Computations have been done using the INRIA Pranas software, B. Cessac et al, Frontiers in Neuroinformatics, Vol 11, page 49, (2017).

<figure><img src="image_64.png" /><figcaption>Figure: Linear response of $f(\omega, n) = \omega_{k_c}(n)$ for different values of stimulus amplitude A.</figcaption></figure>
Computations have been done using the INRIA Pranas software, B. Cessac et al, Frontiers in Neuroinformatics, Vol 11, page 49, (2017).

<figure><img src="image_65.png" /><figcaption>Figure: Linear response of the observable $f(n, \omega) = \omega_{k_c-2}(n-3)\omega_{k_c}(n)$.</figcaption></figure>
Conclusions

* Visual entries are **neither stationary**, nor **spatially regular**.
Conclusions

* Visual entries are **neither stationary**, nor **spatially regular**.

* The « **longitudinal** » pathways PhotoReceptors => Bipolar cells => Ganglion cells efficiently encode the visual information into **parallel flows**.
Conclusions

* Visual entries are **neither stationary**, nor **spatially regular**.

* The « **longitudinal** » pathways PhotoReceptors => Bipolar cells => Ganglion cells efficiently encode the visual information into **parallel flows**.

* On the basis of biophysical modelling + mathematical analysis we found that Amacrine cells may generate, from a **moving stimulus**, **spatio-temporal correlations (waves)** contributing to better encoding a visual scene (e.g. **anticipation**).
Conclusions

* Visual entries are **neither stationary**, nor **spatially regular**.

* The « **longitudinal** » pathways PhotoReceptors => Bipolar cells => Ganglion cells efficiently encode the visual information into **parallel flows**.

* On the basis of biophysical modelling + mathematical analysis we found that Amacrine cells may generate, from a **moving stimulus**, **spatio-temporal correlations (waves)** contributing to better encoding a visual scene (e.g. **anticipation**).

* We also showed theoretically that an object moving in the receptive field of laterally connected Ganglion Cells may dramatically imprint **spike statistics**, not only firing rates, but also **higher order correlations**, living, in the **raster plots**, a trace that could be deciphered.
Conclusions

* Visual entries are **neither stationary**, nor **spatially regular**.

* The « **longitudinal** » pathways PhotoReceptors => Bipolar cells => Ganglion cells efficiently encode the visual information into **parallel flows**.

* On the basis of biophysical modelling + mathematical analysis we found that Amacrine cells may generate, from a **moving stimulus**, **spatio-temporal correlations (waves)** contributing to better encoding a visual scene (e.g. **anticipation**).

* We also showed theoretically that an object moving in the receptive field of laterally connected Ganglion Cells may dramatically imprint **spike statistics**, not only firing rates, but also **higher order correlations**, living, in the **raster plots**, a trace that could be deciphered.

* This leads to a paradigm very different from the « **efficient coding** » paradigm where longitudinal pathways are assumed to be independent. We switch from a paradigm of « rate » decoding to « **correlations** » **decoding**.
Conclusions

* Visual entries are neither stationary, nor spatially regular.

* The « longitudinal » pathways PhotoReceptors => Bipolar cells => Ganglion cells
efficiently encode the visual information into parallel flows.

* On the basis of biophysical modelling + mathematical analysis we found that
Amacrine cells may participate in the informational structure of the
**correlations (waves)** contributing to better encoding a visual scene (e.g.
anticipation).

**How could the visual cortex**
**« decode »** such signals ?

* We also show theoretically that an object moving in the receptive field of laterally connected Ganglion Cells dramatically imprints spike statistics, not only firing rates, but also higher order correlations, living, in the raster plots, a trace that could be deciphered.

* This leads to a paradigm very different from the « efficient coding » paradigm where longitudinal pathways are assumed to be independent. We switch from a paradigm of « rate » decoding to « correlations » decoding.
Thanks 😊