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

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# The early visual system from a dynamical system perspective: Main course

Bruno Cessac

► To cite this version:

Bruno Cessac. The early visual system from a dynamical system perspective: Main course. Presentation at the Latin American Summer School in Computational Neuroscience (LACONEU), Valparaiso, Chile, Jan 2025, Valparaiso (Chile), Chile. hal-04920280

HAL Id: hal-04920280

https://inria.hal.science/hal-04920280v1

Submitted on 30 Jan 2025

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Distributed under a Creative Commons Attribution 4.0 International License
# The early visual system from a dynamical system perspective

B. Cessac, Université Côte d'Azur, INRIA, Biovision team and Neuromod Institute
Sophia Antipolis, France
<figure><img src="image_3.png" /></figure>
Preamble

<figure><img src="image_4.png" /></figure>

Source : http://fr.gde-fon.com/

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

Source : https://theconversation.com/
Preamble

<figure><img src="image_6.png" /></figure>

The external world sends us
photons coming from
complex, multiscale, scenes
with moving objects.

Source : http://fr.gde-fon.com/

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

Our brain is working with
weak electrical currents
generated by ions transfer
(neurons, synapses, glial
cells).

Source : https://theconversation.com/
Preamble

The retina is the interface between these two worlds.

Moving landscapes made of photons are converted into dynamical landscape made of electric potentials.

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

The external world sends us photons coming from complex, multiscale, scenes with moving objects.

Source : http://fr.gde-fon.com/

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

Our brain is working with weak electrical currents generated by ions transfer (neurons, synapses, glial cells).

Source : https://theconversation.com/
# The visual system

<figure><img src="image_10.png" /></figure>
# The visual system

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

Visual
cortex

Retina

LGN

Retina

<figure><img src="image_12.png" /></figure>
# The visual system

The retina is a small membrane, at the back of the eye, with a thickness between 0.1 and 0.4 mm, and an area of about one centimeter square (human).

It contains millions of neurons and synapses.

It is highly structured with specific neurons types organized into specific structures, circuits.

The retina is neither a « camera » nor a « sensor », it is quite smarter than that.

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

Visual cortex

Retina

LGN

<figure><img src="image_14.png" /></figure>
# The visual system

The retina is a small membrane, at the back of the eye, with a thickness between 0.1 and 0.4 mm, and an area of about one centimeter square (human).

It contains millions of neurons and synapses.

It is highly structured with specific neurons types organized into specific structures, circuits.

The retina is neither a « camera » nor a « sensor », it is quite smarter than that.

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

Visual cortex

Retina

LGN

<figure><img src="image_16.png" /></figure>
# A few physical constraints shaping the structure and dynamics of the retina
The retina is the interface between two worlds.
Moving landscapes made of photons are converted into dynamical
landscape made of electric potentials.

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

Neuron : https://www.verywellhealth.com/
The retina is the interface between two worlds.
Moving landscapes made of photons are converted into dynamical
lanscape made of electric potentials.

(1) Cells contain water => currents are made by ions transfer (Na+, K+, Cl-, Ca++, ...).

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

<figure><img src="image_19.png" /></figure>
The retina is the interface between two worlds.
Moving landscapes made of photons are converted into dynamical
lanscape made of electric potentials.

(1) Cells contain water => currents are made by ions transfer (Na+, K+, Cl-, Ca++, ...).

Ion channels are very efficient. A single Na
channel is able to convey from 10⁶ to 10⁸ ions
per second. This corresponds to a current of
0.16 – 16 pA.

<figure><img src="image_20.png" /></figure>
The retina is the interface between two worlds.
Moving landscapes made of photons are converted into dynamical
lanscape made of electric potentials.

(1) Cells contain water => currents are made by ions transfer (Na+, K+, Cl-, Ca++, ...).

(2) The body has to maintain an almost constant temperature => electric currents are weak (~0.1, 1 nA).

Ion channels are very efficient. A single Na
channel is able to convey from 10⁶ to 10⁸ ions
per second. This corresponds to a current of
0.16 – 16 pA.

<figure><img src="image_21.png" /></figure>
(1) Cells contain water => currents are made by ions transfer (Na+, K+, Cl-, Ca++, ...).

(2) The body has to maintain an almost constant temperature => electric currents are weak (~0.1, 1 nA).

(3) So weak electric current could not propagate over distances larger than ~ mm if they were due to passive conduction

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

<figure><img src="image_23.png" /></figure>

NEUROSCIENCE, Third Edition, Figure 3.10 (Part 1) ©2004 Sauer Associates, Inc.

<figure><img src="image_24.png" /></figure>

PURDUE
UNIVERSITY
The retina is the interface between two worlds. Moving landscapes made of photons are converted into dynamical landscape made of electric potentials.

(1) Cells contain water => currents are made by ions transfer (Na+, K+, Cl-, Ca++, ...).

(2) The body has to maintain an almost constant temperature => electric currents are weak (~0.1, 1 nA).

(3) So weak electric current could not propagate over distances larger than ~ mm if they were due to passive conduction => neurons produce action potentials.

<figure><img src="image_25.png" /></figure>

<figure><img src="image_26.png" /></figure>

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

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

Source: http://what-when-how.com/neuroscience/electrophysiology-of-neurons-the-neuron-part-3/
The retina is the interface between two worlds.
Moving landscapes made of photons are converted into dynamical
lanscape made of electric potentials.

(1) Cells contain water => currents are made by ions transfer (Na+, K+, Cl-, Ca++, ...).

(2) The body has to maintain an almost constant temperature => electric currents are weak (~0.1, 1 nA).

(3) So weak electric current could not propagate over distances larger than ~ mm if they were due to passive conduction => neurons produce action potentials.

(4) Ions are slow => Massively parallel
« computation »

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

Source: https://levekunst.com/hacking-the-self/

Neurons make networks. They are connected by synapses (and glial cells). « Computations » are essentially made by the synapses.
The retina is the interface between two worlds.
Moving landscapes made of photons are converted into dynamical
lanscape made of electric potentials.

(1) Cells contain water => currents are made by ions transfer (Na+, K+, Cl-, Ca++, ...).

(2) The body has to maintain an almost constant temperature => electric currents are weak (~0.1, 1 nA).

(3) So weak electric current could not propagate over distances larger than ~ mm if they were due to passive conduction => neurons produce action potentials.

(4) Ions are slow => Massively parallel
« computation »

(5) The retina has to capture photons in the visible range => converted into electric potentials.

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

Source :https://medlineplus.gov/

Photoreceptors are very efficient « light converters ».
The retina is the interface between two worlds.
Moving landscapes made of photons are converted into dynamical
lanscape made of electric potentials.

(1) Cells contain water => currents are made by ions transfer (Na+, K+, Cl-, Ca++, ...).

(2) The body has to maintain an almost constant temperature => electric currents are weak (~0.1, 1 nA).

(3) So weak electric current could not propagate over distances larger than ~ mm if they were due to passive conduction => neurons produce action potentials.

(4) Ions are slow => Massively parallel
« computation »

(5) The retina has to capture photons in the visible range => converted into electric potentials.

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

Source :https://medlineplus.gov/

Photoreceptors are very efficient « light converters ». 1 rhodopsin molecule activated => 1 mV change in the membrane potential.
The visual system

<figure><img src="image_32.png" /></figure>
# The visual system

<figure><img src="image_33.png" /><figcaption>## Retina's layered structure</figcaption></figure>

<figure><img src="image_34.png" /></figure>
# The visual system

## Retina's layered structure

<figure><img src="image_35.png" /></figure>
# The visual system

<figure><img src="image_36.png" /></figure>

## Retina's layered structure
# The visual system

## Retina's layered structure

<figure><img src="image_37.png" /></figure>

« Information » flux
# The visual system

## Retina's layered structure

« Encoder »

Light

<figure><img src="image_38.png" /></figure>

nghion cells bipolar cells photoreceptors

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

amacrine cells horizontal cells

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

« Decoder »

Visual cortex

LGN

« Information » flux

<figure><img src="image_41.png" /></figure>
# The visual system

## Retina's layered structure

« Encoder »

nghion cells bipolar cells photoreceptors

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

amacrine cells

horizontal cells

<figure><img src="image_43.png" /></figure>

« Decoder »

<figure><img src="image_44.png" /></figure>

« Information » flux

<figure><img src="image_45.png" /></figure>
The retina is the interface between two worlds.
Moving landscapes made of photons are converted into dynamical
lanscape made of electric potentials.

(1) Cells contain water => currents are made by ions transfer (Na+, K+, Cl-, Ca++, ...).

(2) The body has to maintain an almost constant
temperature => electric currents are weak
(~ 0.1, 1 nA).

(3) So weak electric current could not propagate
over distances larger than ~ mm if they were
due to passive conduction => neurons produce
action potentials.

(4) Ions are slow => Massively parallel
« computation »

(5) The retina has to capture photons in the visible range => converted into electric potentials.
The retina is the interface between two worlds.
Moving landscapes made of photons are converted into dynamical
lanscape made of electric potentials.

(1) Cells contain water => currents are made by ions transfer (Na+, K+, Cl-, Ca++, ...).

(2) The body has to maintain an almost constant
temperature => electric currents are weak
(~ 0 .1, 1 nA).

(3) So weak electric current could not propagate
over distances larger than ~ mm if they were
due to passive conduction => neurons produce
action potentials.

(4) Ions are slow => Massively parallel
« computation »

(5) The retina has to capture photons in the visible range => converted into electric potentials.

(6) **The retina is parcimonious => Transmits only**
**what is « informative »**.
The retina is the interface between two worlds.
Moving landscapes made of photons are converted into dynamical
lanscape made of electric potentials.

(1) Cells contain water => currents are made by ions transfer (Na+, K+, Cl-, Ca++, ...).

(2) The body has to maintain an almost constant temperature => electric currents are weak (~0.1, 1 nA).

(3) So weak electric current could not propagate over distances larger than ~ mm if they were due to passive conduction => neurons produce action potentials.

(4) Ions are slow => Massively parallel
« computation »

(5) The retina has to capture photons in the visible range => converted into electric potentials.

(6) **The retina is parcimonious => Transmits only what is « informative »**.

<figure><img src="image_46.png" /><figcaption>Examples of saccadic eye movements from Yarbus
(1967)</figcaption></figure>
The retina is the interface between two worlds.
Moving landscapes made of photons are converted into dynamical
landscape made of electric potentials.

(1) Cells contain water => currents are made by ions transfer (Na+, K+, Cl-, Ca++, ...).

(2) The body has to maintain an almost constant temperature => electric currents are weak (~0.1, 1 nA).

(3) So weak electric current could not propagate over distances larger than ~ mm if they were due to passive conduction => neurons produce action potentials.

(4) Ions are slow => Massively parallel
« computation »

(5) The retina has to capture photons in the visible range => converted into electric potentials.

(6) **The retina is parcimonious => Transmits only what is « informative ».**

<figure><img src="image_47.png" /><figcaption>Examples of saccadic eye movements from Yarbus
(1967)</figcaption></figure>

<figure><img src="image_48.png" /></figure>
How do our eyes learn to see ?
# The structure of the retina during development

Retina's layered structure is shaped during development

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

ganglion cells

bipolar cells

photoreceptors

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

amacrine cells

horizontal cells

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

Visual cortex

LGN

But How?

<figure><img src="image_52.png" /></figure>

Retinal waves!
# Retinal waves

Spontaneous spatio-temporal
waves during development
 disappear short after birth
when vision is functional

Recordings from the retina

<figure><img src="image_53.png" /></figure>

<figure><img src="image_54.png" /></figure>

Multi-electrode
array (MEA)

<figure><img src="image_55.png" /></figure>

MEA recording of the voltage from
a P11 mouse retina in the presence
of 10 µM bicuculline.
# Stages of Retinal Waves During Development

<figure><img src="image_56.png" /></figure>

* Formation of retina circuitry

* Chemical synapses *not* formed yet

* Gap junction-mediated
# Retinotopy

<figure><img src="image_57.png" /><figcaption>Figure 2.3: Cartoon showing the retinotopic projection and relationship with the visual field. Here a cat is presented as an image to a mouse's retina. This image is inverted by the lens of the eye and projected topographically into the SC where the neighbour relationships can be traced using the colour gradient. Figure adapted from Seabrook et. al. (2017) [190].</figcaption></figure>
Retinotopy

Retinotopy results from :

1) **Competition** at the synaptic level (*fight for resources*)

2) **Chemotaxis** (retinal afferents affinity to colliculus location depends on the concentration of ligands and receptors at the location)

3) **Activity** (effects of neuronal activity on the organisation of topography)

4) **Plasticity**
# Retinotopy

Retinotopy results from :

1) **Competition** at the synaptic level (*fight for resources*)

2) **Chemotaxis** (retinal afferents affinity to colliculus location depends on the *concentration of ligands and receptors* at the location)

3) **Activity** (effects of *neuronal activity* on the organisation of topography)

4) **Plasticity**

<figure><img src="image_58.png" /></figure>

Variational
formulation
# Retinotopy

Retinotopy results from :

1) **Competition** at the synaptic level (*fight for resources*)

2) **Chemotaxis** (retinal afferents affinity to colliculus location depends on the *concentration of ligands and receptors* at the location)

3) **Activity** (effects of *neuronal activity* on the organisation of topography)

4) **Plasticity**

+

<figure><img src="image_59.png" /></figure>

## Retinal waves input

From Nicholas Gale thesis, «Theoretical Investigations into Principles of Topographic Map Formation and Applications», Cambridge 2022.
# The Starburst Amacrine Cells Network

## Model synaptic interactions

<figure><img src="image_60.png" /></figure>

SACs realistic connections

SACs on a lattice

SACs become points on a lattice
A network model for stage II retinal waves

D. Matzakou-Karvouniari, L. Gil, E. Orendorff, O. Marre, S. Picaud, B. Cessac, « A biophysical model explains the spontaneous bursting behavior in the developing retina », Scientific Reports, Nature Publishing Group, 2019, 9 (1), pp.1-23. (10.1038/s41598-018-38299-4)

$$
\left\{
\begin{aligned}
C_m \frac{dV_i}{dt} &= -g_L(V_i - V_L) - g_C M_\infty(V_i)(V_i - V_C) - g_K N_i(V_i - V_K) \\
&\quad - G_{sAHP}(R_i)(V_i - V_K) - G_A(\{A_k\}_{k \in B_i})(V_i - V_A) \\
\tau_N \frac{dN_i}{dt} &= \Lambda(V_i)(N_\infty(V_i) - N_i) \\
\tau_C \frac{dC_i}{dt} &= -\frac{\alpha_C}{H_X} C_i + C_0 - \delta_C g_C M_\infty(V_i)(V_i - V_C) \\
\tau_S \frac{dS_i}{dt} &= \alpha_S (1 - S_i) C_i^4 - S_i \\
\tau_R \frac{dR_i}{dt} &= \alpha_R S_i (1 - R_i) - R_i \\
\frac{dA_i}{dt} &= -\mu A_i + \beta_A T_A(V_i),
\end{aligned}
\right.
$$
Fast time scale

Morris-Lecar model with fast dynamics, in the presence of a slow external current

$$ egin{cases} \frac{dV}{dt_f} = -\tilde{g}_L(V - V_L) - \tilde{g}_C M_\infty(V)(V - V_C) - \tilde{g}_K N(V - V_K) + I_{\text{ext}} \\ \\ \frac{dN}{dt_f} = \frac{\tau_L}{\tau_N} \Lambda(V)(N_\infty(V) - N) \end{cases} $$

A

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

Experiment by Zheng et al. 2006,
Nature

B

<figure><img src="image_62.png" /></figure>

Model. Tuning parameters V1, V2,
V3,V4 from ML adjusted to
reproduce the experiment
<figure><img src="image_63.png" /><figcaption>Bifurcation diagram of M-L model</figcaption></figure>
<figure><img src="image_64.png" /><figcaption>Bifurcation diagram of M-L model</figcaption></figure>
Bifurcation diagram of M-L model

<figure><img src="image_65.png" /></figure>
Bifurcation diagram of M-L model

<figure><img src="image_66.png" /></figure>
<figure><img src="image_67.png" /></figure>

saddle-node bifurcation

SN2

SN1

Homoclinic
# Spontaneous Bursting Mechanism of SACs

<figure><img src="image_68.png" /></figure>
The nonlinear dynamics of retinal waves

D. Matzakou-Karvouniari, L. Gil, E. Orendorff, O. Marre, S. Picaud, B. Cessac, « A biophysical model explains the spontaneous bursting behavior in the developing retina », Scientific Reports, Nature Publishing Group, 2019, 9 (1), pp.1-23. (10.1038/s41598-018-38299-4)

B. Cessac, D. Matzakou-Karvouniari, *The non linear dynamics of retinal waves* Physica D: Volume 439, November 2022, 133436 https://hal.inria.fr/hal-03485137/file/main.pdf
The nonlinear dynamics of retinal waves

D. Matzakou-Karvouniari, L. Gil, E. Orendorff, O. Marre, S. Picaud, B. Cessac, « A biophysical model explains the spontaneous bursting behavior in the developing retina », Scientific Reports, Nature Publishing Group, 2019, 9 (1), pp.1-23. (10.1038/s41598-018-38299-4)

B. Cessac, D. Matzakou-Karvouniari, *The non linear dynamics of retinal waves Physica D: Volume 439, November 2022*,
133436 https://hal.inria.fr/hal-03485137/file/main.pdf

• Stage II retinal waves are due to the conjunction of :

1) Spontaneous bursting (initiation) ;

<figure><img src="image_69.png" /></figure>
The nonlinear dynamics of retinal waves

D. Matzakou-Karvouniari, L. Gil, E. Orendorff, O. Marre, S. Picaud, B. Cessac, « A biophysical model explains the spontaneous bursting behavior in the developing retina », Scientific Reports, Nature Publishing Group, 2019, 9 (1), pp.1-23. (10.1038/s41598-018-38299-4)

B. Cessac, D. Matzakou-Karvouniari, *The non linear dynamics of retinal waves Physica D: Volume 439, November 2022*, 133436 https://hal.inria.fr/hal-03485137/file/main.pdf

* Stage II retinal waves are due to the conjunction of :

1) Spontaneous bursting (initiation) ;

2) Acetylcholine network (propagation) ;

<figure><img src="image_70.png" /></figure>
The nonlinear dynamics of retinal waves

D. Matzakou-Karvouniari, L. Gil, E. Orendorff, O. Marre, S. Picaud, B. Cessac, « A biophysical model explains the spontaneous bursting behavior in the developing retina », Scientific Reports, Nature Publishing Group, 2019, 9 (1), pp.1-23. (10.1038/s41598-018-38299-4)

B. Cessac, D. Matzakou-Karvouniari, The non linear dynamics of retinal waves Physica D: Volume 439, November 2022, 133436 https://hal.inria.fr/hal-03485137/file/main.pdf

• Stage II retinal waves are due to the conjunction of :

1) Spontaneous bursting (initiation) ;

2) Acetylcholine network (propagation) ;

3) Long refractoriness (stop);

<figure><img src="image_71.png" /></figure>
The nonlinear dynamics of retinal waves

D. Matzakou-Karvouniari, L. Gil, E. Orendorff, O. Marre, S. Picaud, B. Cessac, « A biophysical model explains the spontaneous bursting behavior in the developing retina », Scientific Reports, Nature Publishing Group, 2019, 9 (1), pp.1-23. (10.1038/s41598-018-38299-4)

B. Cessac, D. Matzakou-Karvouniari, *The non linear dynamics of retinal waves Physica D: Volume 439, November 2022*, 133436 https://hal.inria.fr/hal-03485137/file/main.pdf

• Stage II retinal waves are due to the conjunction of :

1) Spontaneous bursting (initiation) ;

2) Acetylcholine network (propagation) ;

3) Long refractoriness (stop);

<figure><img src="image_72.png" /></figure>

Experiment for coupled and isolated neurons, Zheng et al., 2006, Nature

<figure><img src="image_73.png" /></figure>
# The nonlinear dynamics of retinal waves

D. Matzakou-Karvouniari, L. Gil, E. Orendorff, O. Marre, S. Picaud, B. Cessac, « A biophysical model explains the spontaneous bursting behavior in the developing retina », Scientific Reports, Nature Publishing Group, 2019, 9 (1), pp.1-23. (10.1038/s41598-018-38299-4)

B. Cessac, D. Matzakou-Karvouniari, *The non linear dynamics of retinal waves Physica D: Volume 439, November 2022*, 133436 https://hal.inria.fr/hal-03485137/file/main.pdf

* Stage II retinal waves are due to the conjunction of :

1) Spontaneous bursting (initiation) ;

2) Acetylcholine network (propagation) ;

3) Long refractoriness (stop);

* Waves propagate in a **random landscape** with **refractory zones** left by the previous waves (similar to forest fires although more complex).

<figure><img src="image_74.png" /></figure>

https://webvision.med.utah.edu/book/part-vi-development-of-cell-types-and-synaptic-connections-in-the-retina/formation-of-early-retinal-circuits-in-the-inner-plexiform-layer/
The nonlinear dynamics of retinal waves

D. Matzakou-Karvouniari, L. Gil, E. Orendorff, O. Marre, S. Picaud, B. Cessac, « A biophysical model explains the spontaneous bursting behavior in the developing retina », Scientific Reports, Nature Publishing Group, 2019, 9 (1), pp.1-23. (10.1038/s41598-018-38299-4)

B. Cessac, D. Matzakou-Karvouniari, *The non linear dynamics of retinal waves Physica D: Volume 439, November 2022*,
133436 https://hal.inria.fr/hal-03485137/file/main.pdf

* Stage II retinal waves are due to the conjunction of :

1) Spontaneous bursting (initiation) ;

2) Acetylcholine network (propagation) ;

3) Long refractoriness (stop);

* Waves propagate in a **random landscape** with **refractory zones** left by the previous waves (similar to forest fires although more complex).

* The distribution of waves (long tail) looks like a **critical** phenomena. However, this regime is robust to physiological parameters variation => **Griffiths phase** ?
  (phase transition of infinite order)

<figure><img src="image_75.png" /></figure>

https://phys.org/news/2020-02-electronic-griffiths-phase-solid-state-physical.html
The nonlinear dynamics of retinal waves

D. Matzakou-Karvouniari, L. Gil, E. Orendorff, O. Marre, S. Picaud, B. Cessac, « A biophysical model explains the spontaneous bursting behavior in the developing retina », Scientific Reports, Nature Publishing Group, 2019, 9 (1), pp.1-23. (10.1038/s41598-018-38299-4)

B. Cessac, D. Matzakou-Karvouniari, *The non linear dynamics of retinal waves Physica D: Volume 439, November 2022*,
133436 https://hal.inria.fr/hal-03485137/file/main.pdf

* Stage II retinal waves are due to the conjunction of :

1) Spontaneous bursting (initiation) ;

2) Acetylcholine network (propagation) ;

3) Long refractoriness (stop);

* Waves propagate in a **random landscape** with **refractory zones** left by the previous waves (similar to forest fires although more complex).

* The distribution of waves (long tail) looks like a **critical** phenomena. However, this regime is robust to physiological parameters variation => **Griffiths phase** ?
  (phase transition of infinite order)

* Cf visual cortex. Girardi-Schappo, M., Bortolotto, G., Gonsalves, J. et al. Griffiths phase and long-range correlations in a biologically motivated visual cortex model. Sci Rep 6, 29561 (2016).

<figure><img src="image_76.png" /></figure>
The nonlinear dynamics of retinal waves

D. Matzakou-Karvouniari, L. Gil, E. Orendorff, O. Marre, S. Picaud, B. Cessac, « A biophysical model explains the spontaneous bursting behavior in the developing retina », Scientific Reports, Nature Publishing Group, 2019, 9 (1), pp.1-23. (10.1038/s41598-018-38299-4)

B. Cessac, D. Matzakou-Karvouniari, *The non linear dynamics of retinal waves Physica D: Volume 439, November 2022*, 133436 https://hal.inria.fr/hal-03485137/file/main.pdf

* Stage II retinal waves are due to the conjunction of :

1) Spontaneous bursting (initiation) ;

2) Acetylcholine network (propagation) ;

3) Long refractoriness (stop);

* Waves propagate in a **random landscape** with **refractory zones** left by the previous waves (similar to forest fires although more complex).

* The distribution of waves (long tail) looks like a **critical** phenomena. However, this regime is robust to physiological parameters variation => **Griffiths phase** ? (phase transition of infinite order)

* Cf visual cortex. Girardi-Schappo, M., Bortolotto, G., Gonsalves, J. et al. Griffiths phase and long-range correlations in a biologically motivated visual cortex model. Sci Rep 6, 29561 (2016).

* This complex dynamics can be studied thanks to **bifurcation analysis** and **transport equations**.

<figure><img src="image_77.png" /></figure>

<figure><img src="image_78.png" /></figure>

<figure><img src="image_79.png" /></figure>

<figure><img src="image_80.png" /></figure>

<figure><img src="image_81.png" /></figure>
How do our eyes learn to see ?

• Our eyes start to learn and see **before birth**.

* This process (retinal waves) is **spontaneous** and does not require a huge data basis of « natural » images.

* This process is partly controlled by genetics but **nonlinear dynamics** plays a major role.

• (Partly) due to this mechanism the visual system is able to handle **motion**.
How do our eyes learn to see ?

• Our eyes start to learn and see **before birth**.

* This process (retinal waves) is **spontaneous** and does not require a huge data basis of « natural » images.

* This process is partly controlled by genetics but **nonlinear dynamics** plays a major role.

• (Partly) due to this mechanism the visual system is able to handle **motion**.

Retinal waves as training features spanning the « space » of spatio-temporal percepts ?
# How do our eyes learn to see ?

* Our eyes start to learn and see **before birth**.

* This process (retinal waves) is **spontaneous** and does not require a huge data basis of « natural » images.

* This process is partly controlled by genetics but **nonlinear dynamics** plays a major role.

* (Partly) due to this mechanism the visual system is able to handle **motion**.

**Retinal waves as training features spanning the « space » of spatio-temporal percepts ?**

Waves **distribution** plays a major role
+
dynamical shaping of the retina and visual cortex.
Why can we play tennis ?
# Why can we play tennis ?

Light

<figure><img src="image_82.png" /></figure>

## Retina's layered structure

ganglion cells bipolar cells photoreceptors

<figure><img src="image_83.png" /></figure>

amacrine cells horizontal cells

<figure><img src="image_84.png" /></figure>

Visual cortex

LGN

<figure><img src="image_85.png" /></figure>

Retina

<figure><img src="image_86.png" /></figure>
# Why can we play tennis ?

## Retina's layered structure

<figure><img src="image_87.png" /></figure>
# Why can we play tennis ?

Transmission delay eye-visual cortex~ **100** ms.

Average speed of a tennis ball (stroke) : 170 km/h ~ 47 m/s => **4.7** m in 100 ms

<figure><img src="image_88.png" /></figure>
Motion anticipation

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

<figure><img src="image_89.png" /><figcaption>B</figcaption></figure>

<figure><img src="image_90.png" /></figure>

G. Benvenuti, S. Chemla, A. Boonman, L. Perrinet, G. S. Masson, and F. Chavane. Anticipatory responses along motion trajectories in awake monkey area v1. bioRxiv, 2020.
Anticipation by latency in the primary visual cortex (V1)

VSDI time-course

<figure><img src="image_91.png" /></figure>
Anticipation by latency in the primary visual cortex (V1)

VSDI time-course

<figure><img src="image_92.png" /></figure>
Anticipation by latency in the primary visual cortex (V1)

<figure><img src="image_93.png" /></figure>

G.Benvenuti et al.,
bioRxiv (2020)
Anticipation by latency in the primary visual cortex (V1)

<figure><img src="image_94.png" /></figure>
Anticipation by latency in the primary visual cortex (V1)

<figure><img src="image_95.png" /></figure>

<figure><img src="image_96.png" /></figure>
Anticipation by latency in the primary visual cortex (V1)

<figure><img src="image_97.png" /></figure>

Cortical columns sensing
the bar sends excitation
to others not already
activated by it.
Anticipation by latency in the primary visual cortex (V1)

<figure><img src="image_98.png" /></figure>

Cortical columns sensing the bar sends excitation to others not already activated by it.

<figure><img src="image_99.png" /></figure>

The moving bar elicits anticipatory wave propagating faster than the bar.
Anticipation by latency in the primary visual cortex (V1)

<figure><img src="image_100.png" /></figure>

Cortical columns sensing the bar sends excitation to others not already activated by it.

<figure><img src="image_101.png" /></figure>

The moving bar elicits anticipatory wave propagating faster than the bar.
Anticipation by latency in the primary visual cortex (V1)

<figure><img src="image_102.png" /></figure>

Cortical columns sensing
the bar sends excitation
to others not already
activated by it.

<figure><img src="image_103.png" /></figure>

The moving bar elicits
anticipatory wave
propagating faster than
the bar.
Anticipation by latency in the primary visual cortex (V1)

<figure><img src="image_104.png" /></figure>

Cortical columns sensing the bar sends excitation to others not already activated by it.

<figure><img src="image_105.png" /></figure>

The moving bar elicits anticipatory wave propagating faster than the bar.
Anticipation by latency in the primary visual cortex (V1)

<figure><img src="image_106.png" /></figure>

Cortical columns sensing
the bar sends excitation
to others not already
activated by it.

<figure><img src="image_107.png" /></figure>

The moving bar elicits
anticipatory wave
propagating faster than
the bar.
Anticipation by latency in the primary visual cortex (V1)

<figure><img src="image_108.png" /></figure>

Cortical columns sensing
the bar sends excitation
to others not already
activated by it.

<figure><img src="image_109.png" /></figure>

The moving bar elicits
anticipatory wave
propagating faster than
the bar.
Anticipation by latency in the primary visual cortex (V1)

<figure><img src="image_110.png" /></figure>

Cortical columns sensing
the bar sends excitation
to others not already
activated by it.

<figure><img src="image_111.png" /></figure>

The moving bar elicits
anticipatory wave
propagating faster than
the bar.
Anticipation by latency in the primary visual cortex (V1)

<figure><img src="image_112.png" /></figure>

Cortical columns sensing
the bar sends excitation
to others not already
activated by it.

<figure><img src="image_113.png" /></figure>

The moving bar elicits
anticipatory wave
propagating faster than
the bar.
# A cortical model of anticipation

## Modelling the primary visual cortex network

### Biology

<figure><img src="image_114.png" /></figure>

### Model

Inspired from cortical column organisation.

Y. Zerlaut et al.
Jour. of Comp.
Neurosciences,
(2017)

Di Volo et al.
Neural Comp.,
(2017)

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

J. Emonet, S. Souihel, M. Di Volo, A. Destexhe, F. Chavane, B. Cessac, 2024, submitted
# A cortical model of anticipation

## Modelling the primary visual cortex network

### Biology

<figure><img src="image_115.png" /></figure>

### Model

<figure><img src="image_116.png" /></figure>

Inspired from cortical column organisation.

Y. Zerlaut et al.
Jour. of Comp.
Neurosciences,
(2017)

Di Volo et al.
Neural Comp.,
(2017)

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

J. Emonet, S. Souihel, M. Di Volo, A. Destexhe, F. Chavane, B. Cessac, 2024, submitted
A cortical model of anticipation

$$ \nu_e^{input}(x,t) = \nu_e^{drive} + \int_{\mathbb{R}} dy \mathcal{N}_e(x-y) \nu_e(y, t - \|y-x\|/v_c) $$

$$ \nu_i^{input}(x,t) = \int_{\mathbb{R}} dy \mathcal{N}_i(x-y) \nu_i(y, t - \|y-x\|/v_c) $$

$$ T \frac{\partial \nu_e(x,t)}{\partial t} = - \nu_e(x,t) + $$

$$ \mathcal{F}_e(\nu_e^{aff}(x,t) + \nu_e^{input}(x,t), \nu_i^{input}(x,t)) $$

$$ T \frac{\partial \nu_i(x,t)}{\partial t} = - \nu_i(x,t) + \mathcal{F}_i(\nu_e^{input}(x,t), \nu_i^{input}(x,t)) $$

$$ \mu_V(\nu_e, \nu_i) = \frac{\mu_{Ge} E_e + \mu_{Gi} E_i + g_L E_L}{\mu_G} $$

$$ VSDI = \frac{\mu_V - \mu_{V0}}{\mu_{V0}} $$

Y. Zerlaut et al.
Jour. of Comp.
Neurosciences,
(2017)

Di Volo et al.
Neural Comp.,
(2017)

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

J. Emonet, S. Souihel, M. Di Volo, A. Destexhe, F. Chavane, B.
Cessac, 2024, submitted
A cortical model of anticipation

$$ \nu_e^{\text{input}}(x, t) = \int_{-\|y-x\|/v_c}^{\|y-x\|/v_c} \dots $$

$$ \nu_i^{\text{input}}(x, t) = \int_{-\|x\|/v_c}^{\|x\|/v_c} \dots $$

Excitatory population

$$ T \frac{\partial \nu_e(x, t)}{\partial t} = - \nu_e(x, t) + \mathcal{F}_e(\nu_e^{aff}(x, t) + \nu_e^{\text{input}}(x, t), \nu_i^{\text{input}}(x, t)) $$

$$ T \frac{\partial \nu_i(x, t)}{\partial t} = - \nu_i(x, t) + \mathcal{F}_i(\nu_e^{\text{input}}(x, t), \nu_i^{\text{input}}(x, t)) $$

$$ \mu_V(\nu_e, \nu_i) = \frac{\mu_{Ge} E_e + \mu_{Gi} E_i + g_L E_L}{\mu_G} $$

$$ VSDI = \frac{\mu_V - \mu_{VO}}{\mu_{VO}} $$

Y. Zerlaut et al.
Jour. of Comp.
Neurosciences,
(2017)

Di Volo et al.
Neural Comp.,
(2017)

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

J. Emonet, S. Souihel, M. Di Volo, A. Destexhe, F. Chavane, B.
Cessac, 2024, submitted
A cortical model of anticipation

$$ \nu_e^{\text{input}}(x,t) = \nu_e^{\text{drive}} + \int_{\mathbb{R}} dy \mathcal{N}_e(x-y) \nu_e(y, t - \|y-x\|/v_c) $$

$$ \nu_i^{\text{input}}(x,t) = \int dy \mathcal{N}_i(x-y) \nu_i(y, t - \|y-x\|/v_c) $$

Excitatory input

$$ \nu_e(x,t) + $$

$$ (\nu_e^{aff}(x,t) + \nu_e^{\text{input}}(x,t), \nu_i^{\text{input}}(x,t)) $$

$$ \frac{\partial t}{\partial t} - \nu_i(x,t) + \mathcal{F}_i(\nu_e^{\text{input}}(x,t), \nu_i^{\text{input}}(x,t)) $$

$$ \mu_V(\nu_e, \nu_i) = \frac{\mu_{Ge} E_e + \mu_{Gi} E_i + g_L E_L}{\mu_G} $$

$$ VSDI = \frac{\mu_V - \mu_{V0}}{\mu_{V0}} $$

Y. Zerlaut et al.
Jour. of Comp.
Neurosciences,
(2017)

Di Volo et al.
Neural Comp.,
(2017)

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

J. Emonet, S. Souihel, M. Di Volo, A. Destexhe, F. Chavane, B.
Cessac, 2024, submitted
# A cortical model of anticipation

$$ \nu_e^{input}(x, t) = \nu_e^{drive} + \int_{\mathbb{R}} dy \mathcal{N}_e(x - y) \nu_e(y, t - \|y - x\|/v_c) $$

$$ \nu_i^{input}(x, t) = \int_{\mathbb{R}} dy \mathcal{N}_i(x - y) \nu_i(y, t - \|y - x\|/v_c) $$

$$ T \frac{\partial \nu_e(x, t)}{\partial t} = - \nu_e(x, t) + $$

$$ \nu_e^{aff}(x, t) + \nu_e^{input}(x, t), \nu_i^{input}(x, t) $$

$$ \nu_i(x, t) + \mathcal{F}_i(\nu_e^{input}(x, t), \nu_i^{input}(x, t)) $$

$$ \frac{e E_e + \mu_{Gi} E_i + g_L E_L}{\mu_G} $$

$$ VSDI = \frac{\mu_V - \mu_{V0}}{\mu_{V0}} $$

Inhibitory input

Y. Zerlaut et al.
Jour. of Comp.
Neurosciences,
(2017)

Di Volo et al.
Neural Comp.,
(2017)

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

J. Emonet, S. Souihel, M. Di Volo, A. Destexhe, F. Chavane, B.
Cessac, 2024, submitted
# A cortical model of anticipation

$$ \nu_e^{input}(x, t) = \nu_e^{drive} + \int_{\mathbb{R}} dy \mathcal{N}_e(x - y) \nu_e(y, t - \|y - x\|/v_c) $$

$$ \nu_i^{input}(x, t) = \int_{\mathbb{R}} dy \mathcal{N}_i(x - y) \nu_i(y, t - \|y - x\|/v_c) $$

$$ T \frac{\partial \nu_e(x, t)}{\partial t} = - \nu_e(x, t) + $$

$$ \mathcal{F}_e(\nu_e^{aff}(x, t) + \nu_e^{input}(x, t), \nu_i^{input}(x, t)) $$

$$ T \frac{\partial \nu_i(x, t)}{\partial t} = - \nu_i(x, t) + \mathcal{F}_i(\nu_e^{input}(x, t), \nu_i^{input}(x, t)) $$

$$ \mu_V(\nu_e, \frac{\nu_i E_L}{VSDI}) $$

Thalamic input

Y. Zerlaut et al.
Jour. of Comp.
Neurosciences,
(2017)

Di Volo et al.
Neural Comp.,
(2017)

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

J. Emonet, S. Souihel, M. Di Volo, A. Destexhe, F. Chavane, B.
Cessac, 2024, submitted
A cortical model of anticipation

<figure><img src="image_117.png" /></figure>
A cortical model of anticipation

$$ \nu_e^{\text{input}}(x,t) = \nu_e^{\text{drive}} + \int_{\mathbb{R}} dy \mathcal{N}_e(x-y) \nu_e(y, t - \|y-x\|/v_c) $$

$$ \nu_i^{\text{input}}(x,t) = \int_{\mathbb{R}} dy \mathcal{N}_i(x-y) \nu_i(y, t - \|y-x\|/v_c) $$

$$ T \frac{\partial \nu_e(x,t)}{\partial t} = - \nu_e(x,t) + $$

$$ \mathcal{F}_e(\nu_e^{\text{eff}}(x,t) + \nu_e^{\text{input}}(x,t), \nu_i^{\text{input}}(x,t)) $$

$$ T \frac{\partial \nu_i(x,t)}{\partial t} = - \nu_i(x,t) + \mathcal{F}_i(\nu_i^{\text{input}}(x,t), \nu_i^{\text{input}}(x,t)) $$

$$ \mu_V(\nu_e, \nu_i) = \frac{\mu_{Ge} E_e + \mu_{Gi} E_i + g_L E_L}{\mu_{V0}} $$

$$ VSDI = \frac{\mu_V - \mu_{V0}}{\mu_{V0}} $$

Non linear transfer
functions

Y. Zerlaut et al.
Jour. of Comp.
Neurosciences,
(2017)

Di Volo et al.
Neural Comp.,
(2017)

A cortical model of anticipation in the retina and the
primary visual cortex

J. Emonet, S. Souihel, M. Di Volo, A. Destexhe, F. Chavane, B.
Cessac, 2024, submitted
A cortical model of anticipation

$$ \nu_e^{input}(x,t) = \nu_e^{drive} + \int_{\mathbb{R}} dy \mathcal{N}_e(x-y) \nu_e(y, t - \|y-x\|/v_c) $$

$$ \nu_i^{input}(x,t) = \int_{\mathbb{R}} dy \mathcal{N}_i(x-y) \nu_i(y, t - \|y-x\|/v_c) $$

$$ T \frac{\partial \nu_e(x,t)}{\partial t} = - \nu_e(x,t) + $$

$$ \mathcal{F}_e(\nu_e^{aff}(x,t) + \nu_e^{input}(x,t), \nu_i^{input}(x,t)) $$

$$ T \frac{\partial \nu_i(x,t)}{\partial t} = - \nu_i(x,t) + \mathcal{F}_i(\nu_e^{input}(x,t), \nu_i^{input}(x,t)) $$

$$ \mu_V(\nu_e, \nu_i) = \frac{\mu_{Ge} E_e + \mu_{Gi} E_i + g_L E_L}{VSDI} $$

Mean voltage

Y. Zerlaut et al.
Jour. of Comp.
Neurosciences,
(2017)

Di Volo et al.
Neural Comp.,
(2017)

tion anticipation in the retina and the

primary visual cortex

J. Emonet, S. Souihel, M. Di Volo, A. Destexhe, F. Chavane, B.
Cessac, 2024, submitted
# A cortical model of anticipation

$$ \nu_e^{input}(x, t) = \nu_e^{drive} + \int_{\mathbb{R}} dy \mathcal{N}_e(x - y) \nu_e(y, t - \|y - x\|/v_c) $$

$$ \nu_i^{input}(x, t) = \int_{\mathbb{R}} dy \mathcal{N}_i(x - y) \nu_i(y, t - \|y - x\|/v_c) $$

$$ T \frac{\partial \nu_e(x, t)}{\partial t} = - \nu_e(x, t) + $$

$$ \bar{\mathcal{F}}_e(\nu_e^{aff}(x, t) + \nu_e^{input}(x, t), \nu_i^{input}(x, t)) $$

$$ - \nu_i(x, t) + \mathcal{F}_i(\nu_e^{input}(x, t), \nu_i^{input}(x, t)) $$

$$ \frac{\mu_{Ge} E_e + \mu_{Gi} E_i + g_L E_L}{\mu_G} $$

VSDI signal

$$ VSDI = \frac{\nu - \mu_{VO}}{\mu_{VO}} $$

Y. Zerlaut et al.
Jour. of Comp.
Neurosciences,
(2017)

Di Volo et al.
Neural Comp.,
(2017)

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

J. Emonet, S. Souihel, M. Di Volo, A. Destexhe, F. Chavane, B.
Cessac, 2024, submitted
A cortical model of anticipation

<figure><img src="image_118.png" /></figure>

Y. Zerlaut et al.
Jour. of Comp.
Neurosciences,
(2017)

Di Volo et al.
Neural Comp.,
(2017)

<figure><img src="image_119.png" /></figure>

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

J. Emonet, S. Souihel, M. Di Volo, A. Destexhe, F. Chavane, B. Cessac, 2024, submitted
Retinal Anticipation

**Anticipation** also takes place in the retina

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

Jamie Johnston Leon Lagnado (2015) General features of the retinal connectome determine the computation of motion anticipation, eLife 4:e06250.

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

Michael D. Menz, Dongsoo Lee, and Stephen A. Baccus. Representations of the amacrine cell population underlying retinal motion anticipation. bioRxiv, 2020.
Retinal Anticipation

**Anticipation** also takes place in the retina

<figure><img src="image_120.png" /></figure>

Adapted from M. Menz et al., bioRxiv (2020)
Retinal Anticipation

**Anticipation** also takes place in the retina

<figure><img src="image_121.png" /></figure>

Adapted from M. Menz et al., bioRxiv (2020)
# Retinal Anticipation

Anticipation also takes place in the retina

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

<figure><img src="image_122.png" /></figure>
# Retinal Anticipation

Anticipation also takes place in the retina

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

<figure><img src="image_123.png" /></figure>
# Retinal Anticipation

Anticipation also takes place in the retina

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

* Bipolar voltage :

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

* External drive :

$$ F_{B_i}(t) = \left[ K_i^{S,t} \left( \frac{S}{\tau_B} + \frac{dS}{dt} \right) \right] (t) $$

* Amacrine voltage :

$$ \frac{dV_{A_j}}{dt} = -\frac{1}{\tau_A} V_{A_j}(t) + \sum_{i=1}^{N_A} W_{A_j}^{B_i} R_{B_i}(t). $$

* Coupled dynamics :

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

<figure><img src="image_124.png" /></figure>
# Building a 2D retina model for motion anticipation

## 1) Gain control

How does it work ?

<figure><img src="image_125.png" /></figure>

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

<figure><img src="image_126.png" /></figure>

$$ R_{B_i} = \mathcal{N}_B(V_{B_i}) \mathcal{G}_B(A_{B_i}). $$
Bipolar gain control effect

<figure><img src="image_127.png" /></figure>

J.Emonet et al.
HAL (2024)

<figure><img src="image_128.png" /></figure>

Bipolar gain control changes the shape of the response
in retina and cortex.
Bipolar gain control effect

<figure><img src="image_129.png" /></figure>

J.Emonet et al.
HAL (2024)

<figure><img src="image_130.png" /></figure>

Bipolar gain control shift retinal and cortical peaks earlier
with a preferential intensity where the cortical
peak is in advance over the retinal one.
Bipolar gain control effect

<figure><img src="image_131.png" /></figure>

<figure><img src="image_132.png" /></figure>

Bipolar gain control causes adapting anticipation in the cortex without affecting too much cortical anticipation.
Ganglion gain control effect

<figure><img src="image_133.png" /></figure>

<figure><img src="image_134.png" /></figure>

<figure><img src="image_135.png" /></figure>

Ganglion gain control shifts the peak.
# Retinal Anticipation

## Amacrine cells connectivity

<figure><img src="image_136.png" /></figure>

Ganglion cells are **not** independent encoders
# Retinal Anticipation

<figure><img src="image_137.png" /></figure>

## Local circuit
# Retinal Anticipation

<figure><img src="image_138.png" /><figcaption>Local circuit</figcaption></figure>

<figure><img src="image_139.png" /><figcaption>Network</figcaption></figure>
# Does amacrine biphasic connectivity create predicting anticipation?

## Biology

<figure><img src="image_140.png" /></figure>

Menz et al., bioRxiv
(2020)

## Simulation

<figure><img src="image_141.png" /></figure>

J.Emonet
Thesis (2024)
Retinal Anticipation

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

<figure><img src="image_142.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>
Effect of retinal output amplitude

<figure><img src="image_143.png" /></figure>

<figure><img src="image_144.png" /></figure>

<figure><img src="image_145.png" /></figure>

<figure><img src="image_146.png" /></figure>

The retinal output amplitude accentuates anticipation by latency.
<figure><img src="image_147.png" /></figure>

<figure><img src="image_148.png" /></figure>

4.5°/s

<figure><img src="image_149.png" /></figure>

30°/s

Delay (ms)

<figure><img src="image_150.png" /></figure>

Horizontal space (degree)

Effect of bar speed

<figure><img src="image_151.png" /></figure>

<figure><img src="image_152.png" /></figure>

4.5°/s

Delay (ms)

<figure><img src="image_153.png" /></figure>

30°/s

Delay (ms)

<figure><img src="image_154.png" /></figure>

The speed of the bar reduces anticipation by latency indicators.
<figure><img src="image_155.png" /><figcaption>Effect of cortical extent length</figcaption></figure>

The extension of cortical extensions increases latency
cortical anticipation.

<figure><img src="image_156.png" /></figure>
Feedback inhibition effect

<figure><img src="image_157.png" /></figure>

<figure><img src="image_158.png" /></figure>

<figure><img src="image_159.png" /></figure>

Amacrine feedback inhibition generates adapting anticipation in the cortex with an impact on cortical anticipation.
Feedforward inhibition effect

<figure><img src="image_160.png" /></figure>

<figure><img src="image_161.png" /></figure>

<figure><img src="image_162.png" /></figure>

<figure><img src="image_163.png" /></figure>

<figure><img src="image_164.png" /></figure>

Feedforward inhibition shifts the retinal peak while reducing cortical anticipation by latency more than any other effect.
Retinal Anticipation

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

* A **moving object** induces a **retinal wave ahead** of the stimulus thanks to gain control and the amacrine cells network.

1) Compensation of the **photo-transduction delay**.

2) Simple implementation, **weak consumption**.

3) Most of the « computation » is **analogic**.

* Extend to differential motion, interrupted motion, surprise.
Retinal Anticipation

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

* A **moving object** induces a **retinal wave ahead** of the stimulus thanks to gain control and the amacrine cells network.

1) Compensation of the **photo-transduction delay**.

2) Simple implementation, **weak consumption**.

3) Most of the « computation » is **analogic**.

* Extend to differential motion, interrupted motion, surprise.

**Motion is not a succession of images.**

**There is a spatio-temporal coherence-correlation, captured by stimulus induced activity waves.**
Conclusions

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

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

* The « **longitudinal** » retinal 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** » retinal 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** » retinal 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**).

> Waves could be the fundamental key to understand the visual processing.

> How does the brain efficiently decode a flux of spikes with multi scale spatio-temporal correlations ?
Thanks