# The retinal network from a dynamical system perspective

Bruno Cessac

► To cite this version:

Bruno Cessac. The retinal network from a dynamical system perspective. 2026 - Séminaire de l'Hertie Institute for AI in Brain Health, Jan 2026, Tuebingen, Germany. hal-05479606

HAL Id: hal-05479606

https://inria.hal.science/hal-05479606v1

Submitted on 27 Jan 2026

HAL is a multi-disciplinary open access
archive for the deposit and dissemination of sci-
entific research documents, whether they are pub-
lished or not. The documents may come from
teaching and research institutions in France or
abroad, or from public or private research centers.

L'archive ouverte pluridisciplinaire HAL, est
destinée au dépôt et à la diffusion de documents
scientifiques de niveau recherche, publiés ou non,
émanant des établissements d'enseignement et de
recherche français ou étrangers, des laboratoires
publics ou privés.

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

Distributed under a Creative Commons CC BY 4.0 - Attribution - International License
# The retinal network from a dynamical system perspective

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

The retina is the entrance of the visual system, able to convert the photons from the external world (in the visible range) into electric pulses (spikes) transmitted to the brain via the optic nerve.

The retina contains millions of neurons and synapses. It is highly structured with specific neurons types organized into specific structures, circuits. It is quite efficient in processing motion.

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

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

Visual cortex

Retinal structure

LGN

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

B. Section through retina
# The visual system

## Retina's layered structure

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

« Information » flux
# The visual system

## Retina's layered structure

« Encoder »

nghion cells bipolar cells photoreceptors

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

amacrine cells

horizontal cells

« Decoder »

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

Visual
cortex

LGN

« Information » flux

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

Light

<figure><img src="image_8.png" /></figure>
# The visual system

## Retina's layered structure

« Encoder »

ganglion cells bipolar cells photoreceptors

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

amacrine cells

horizontal cells

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

« Decoder »

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

Visual cortex

LGN

<figure><img src="image_12.png" /></figure>

« Information » flux

<figure><img src="image_13.png" /></figure>
# The visual system

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

There is an evident information flow, from photoreceptors, to bipolar cells, to ganglion cells.

« Information » flux
# The visual system

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

There is an evident information flow, from photoreceptors, to bipolar cells, to ganglion cells.

What is the role of lateral interneurons, horizontal cells and amacrine cells ?

« Information » flux
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).
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**.
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.
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. Which role does **lateral connectivity** play in shaping spatio-temporal patterns in response to visual stimuli ?
A modeller point of view

Based on a **dynamical system** perspective unravel potential **generic** mechanisms, involving **a few** phenomenological parameters, allowing to understand how **lateral connectivity** could shape the retinal **dynamics** in some **specific** examples.
Developmental retinal waves
# The structure of the retina during development

Retina's layered structure is shaped during development

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

ganglion cells

bipolar cells

photoreceptors

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

amacrine cells

horizontal cells

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

Visual cortex

LGN

But How?

<figure><img src="image_19.png" /></figure>

Retinal waves!
Spontaneous spatio-temporal
waves during development

disappear short after birth
 when vision is functional

Recordings from the retina

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

amacrine cells

horizontal cells

<figure><img src="image_21.png" /></figure>

<figure><img src="image_22.png" /><figcaption>Multi-electrode
array (MEA)</figcaption></figure>

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

MEA recording of the voltage from a mouse retina
https://www.youtube.com/watch?v=0kt5_mhTy4w.
# Stages of Retinal Waves During Development

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

* Formation of retina circuitry

* Chemical synapses not formed yet

* Gap junction-mediated
Variability and robustness of
retinal waves

i) Across species

Godfrey et al. 2007

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

<figure><img src="image_26.png" /></figure>
# Variability and robustness of retinal waves

## i) Across species

Godfrey et al. 2007

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

## ii) Development

Degradation of cholinergic nicotinic receptors upon maturation

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

Zheng et al. 2004
# Variability and robustness of retinal waves

## i) Across species

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

Godfrey et al. 2007

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

## ii) Development

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

## iii) Pharmacology

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

P4

P8
# Variability and robustness of retinal waves

## i) Across species

Godfrey et al. 2007

<figure><img src="image_33.png" /></figure>

## ii) Development

Degradation of cholinergic nicotinic receptors upon maturation

<figure><img src="image_34.png" /></figure>

## iii) Pharmacology

<figure><img src="image_35.png" /></figure>

P4

P8

## iv) Spatial Variability

<figure><img src="image_36.png" /></figure>
# Variability and robustness of retinal waves

## i) Across species

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

Godfrey et al. 2007

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

Period (s)

Speed (µm/s)

Rabbit

Tuna

Mice

Cock

What are the sources of these variabilities?

## ii) Development

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

Degradation of cholinergic nicotinic receptors upon maturation

Zheng et al. 2004

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

Parinatabage (d)

## iii) Pharmacology

<figure><img src="image_41.png" /></figure>

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

## iv) Spatial Variability

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

P4

P8
# Focus on stage II retinal waves

*Stage II retinal waves are well studied experimentally and there is already existing work on their modeling*

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

Zheng et al. 2006
# Experiments for the emergence of retinal waves

Retinal waves require three components:

i) Spontaneous bursting activity (waves start)

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

Experiment for isolated neurons,
Zheng et al., 2006, Nature
# Experiments for the emergence of retinal waves

Retinal waves require three components:

i) Spontaneous bursting activity (waves start)

ii) Refractory mechanism (slow After HyperPolarisation- sAHP) (waves stop)

<figure><img src="image_46.png" /><figcaption>Experiment for isolated neurons,
Zheng et al., 2006, Nature</figcaption></figure>
# Experiments for the emergence of retinal waves

Retinal waves require three components:

i) Spontaneous bursting activity (waves start)

ii) Refractory mechanism (slow After HyperPolarisation- sAHP) (waves stop)

iii) Coupling (through Acetylcholine neurotransmitter) (waves propagate)

<figure><img src="image_47.png" /><figcaption>Experiment for coupled and isolated neurons, Zheng et al., 2006, Nature</figcaption></figure>
# Experiments for the emergence of retinal waves

Retinal waves require three components:

i) Spontaneous bursting activity (waves start)

ii) Refractory mechanism (slow After HyperPolarisation- sAHP) (waves stop)

iii) Coupling (through Acetylcholine neurotransmitter) (waves propagate)

<figure><img src="image_48.png" /><figcaption>Experiment for coupled and isolated neurons, Zheng et al., 2006, Nature</figcaption></figure>
# Spontaneous Bursting Mechanism of SACs

<figure><img src="image_49.png" /></figure>
# The Starburst Amacrine Cells Network

## Model synaptic interactions

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

SACs realistic connections

SACs on a lattice

SACs become points on a lattice
Several models of stage II retinal waves have been already proposed :

ii) Refractory mechanism (slow After HyperPolarisation- sAHP)

iii) Coupling (through Acetylcholine neurotransmitter)

• Feller et all Neuron, 1997.

• Eglen et al, J. Neuroscience 2000.

• Godfrey & Swindale. PLoS Comput Biol, 2007.

• Hennig et al, J. Neuroscience, 2009

• Lansdell et al, PLoS Computational Biology, 2014.

But none of them as studied simultaneously these 3 aspects:

(i) realistic model of spontaneous bursting

(ii) sAHP biophysical modelling

(iii) realistic Ach interaction.

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

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

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

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

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

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

<figure><img src="image_56.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

* **Region A-B**: Rest state.

* **Region C**: Fast oscillations.

* **Region D**: Bi-stability: Rest state (low voltage) + fast oscillations (high voltage).

Dynamics returns recurrently in region D.

<figure><img src="image_57.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

* **Region A-B**: Rest state.

* **Region C**: Fast oscillations.

* **Region D**: Bi-stability: Rest state (low voltage) + fast oscillations (high voltage).

*Dynamics returns recurrently in region D.*

1. In region D, a transition from rest state to fast oscillations can occur due to noise (spontaneous ignition).

<figure><img src="image_58.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

* **Region A-B**: Rest state.

* **Region C**: Fast oscillations.

* **Region D**: Bi-stability: Rest state (low voltage) + fast oscillations (high voltage).

*Dynamics returns recurrently in region D.*

1) In region D, a transition from rest state to fast oscillations can occur due to noise (spontaneous ignition).

z

<figure><img src="image_59.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

* **Region A-B**: Rest state.

* **Region C**: Fast oscillations.

* **Region D**: Bi-stability: Rest state (low voltage) + fast oscillations (high voltage).

*Dynamics returns recurrently in region D.*

1. In region D, a transition from rest state to fast oscillations can occur due to noise (**spontaneous ignition**).

2. When a SAC is embedded in a retinal wave, its Ach conductance increases (region C).

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

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

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

<figure><img src="image_63.png" /></figure>

<figure><img src="image_64.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

* **Region A-B**: Rest state.

* **Region C**: Fast oscillations.

* **Region D**: Bi-stability: Rest state (low voltage) + fast oscillations (high voltage).

*Dynamics returns recurrently in region D.*

1. In region D, a transition from rest state to fast oscillations can occur due to noise **(spontaneous ignition)**.

2. When a SAC is embedded in a retinal wave, its Ach conductance increases (region C).

<figure><img src="image_65.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

* **Region A-B**: Rest state.

* **Region C**: Fast oscillations.

* **Region D**: Bi-stability: Rest state (low voltage) + fast oscillations (high voltage).

*Dynamics returns recurrently in region D.*

1. In region D, a transition from rest state to fast oscillations can occur due to noise **(spontaneous ignition)**.

2. When a SAC is embedded in a retinal wave, its Ach conductance increases (region C). There is a nonlinear coupling between bursting SACs, prolongating their burst.

z

<figure><img src="image_66.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

* **Region A-B**: Rest state.

* **Region C**: Fast oscillations.

* **Region D**: Bi-stability : Rest state (low voltage) + fast oscillations (high voltage).

Dynamics returns recurrently in region D.

1) In region D, a transition from rest state to fast oscillations can occur due to noise (spontaneous ignition).

2) When a SAC is embedded in a retinal wave, its Ach conductance increases (region C). There is a nonlinear coupling between bursting SACs, prolongating their burst.

3) Then calcium/sAHP dynamics increases sAHP conductance => goes to region B => End of bursting => rest state with low voltage.

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

<figure><img src="image_68.png" /></figure>

<figure><img src="image_69.png" /></figure>

<figure><img src="image_70.png" /></figure>

<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

* **Region A-B**: Rest state.

* **Region C**: Fast oscillations.

* **Region D**: Bi-stability: Rest state (low voltage) + fast oscillations (high voltage).

Dynamics returns recurrently in region D.

1) In region D, a transition from rest state to fast oscillations can occur due to noise (spontaneous ignition).

2) When a SAC is embedded in a retinal wave, its Ach conductance increases (region C). There is a nonlinear coupling between bursting SACs, prolongating their burst.

3) Then calcium/sAHP dynamics increases sAHP conductance => goes to region B => End of bursting => rest state with low voltage.

<figure><img src="image_72.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

* **Region A-B :** Rest state.

* **Region C :** Fast oscillations.

* **Region D :** Bi-stability : Rest state (low voltage) + fast oscillations (high voltage).

*Dynamics returns recurrently in region D.*

1) In region D, a transition from rest state to fast oscillations can occur due to noise (**spontaneous ignition**).

2) When a SAC is embedded in a retinal wave, its Ach conductance increases (region C). There is a nonlinear coupling between bursting SACs, prolongating their burst.

3) Then calcium/sAHP dynamics increases sAHP conductance => goes to region B => End of bursting => rest state with low voltage.

4) sAHP conductance decreases as well as Ach conductance (neighbours eventually stop bursting too) => return to region D, via region A.

*SAC is ready to burst again.*

<figure><img src="image_73.png" /></figure>

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

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

<figure><img src="image_76.png" /></figure>

<figure><img src="image_77.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_78.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);

* Spontaneous bursting occurs with a probability very sensitive to the position in region D.

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

<figure><img src="image_80.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);

* Spontaneous bursting occurs with a **probability very sensitive** to the position in region D.

* Waves propagate in a **random landscape** with **refractory zones** left by the previous waves, similar to forest fires although more complex, due to **the nonlinear coupling** between bursting cells.

<figure><img src="image_81.png" /></figure>
The nonlinear dynamics of retinal waves

D. Matzakou-Karvouniari, L. Gil, E. Orendorff, O. Marre, S. Picaud,  
spontaneous bursting behavior in the developing retina », Scientific Re  
23. (10.1038/s41598-018-38

B. Cessac, D. Matzakou-Karvouniari, *The non linear dynamics of retina*
133436 https://hal.inria.fr/hal-034851

ii) Development

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

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

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

1) Spontaneous bursting (initiation) ;

2) Acetylcholine network (propagation) ;

3) Long refractoriness (stop);

* Spontaneous bursting occurs with a probability very sensitive to the position in region D.

* Waves propagate in a **random landscape** with **refractory zones** left by the previous waves, similar to forest fires although more complex, due to **the nonlinear coupling** between bursting cells.

* As a consequence, depending on the **maximal Ach conductance**, stage II retinal waves exhibit **different regimes of propagation** (anomalous diffusion, ballistic). The **distribution of sizes and durations varies too**, with a power law regime.
# Variability within retinal waves

## i) Across species

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

Godfrey et al. 2007

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

## ii) Development

<figure><img src="image_86.png" /></figure>

Degradation of cholinergic nicotinic receptors upon maturation

Zheng et al. 2004

## iii) Pharmacology

<figure><img src="image_87.png" /></figure>

P4

P8

## iv) Spatial Variability

<figure><img src="image_88.png" /></figure>
# Variability within retinal waves

## i) Across species

Godfrey et al. 2007

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

## ii) Development

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

Degradation of cholinergic nicotinic
receptors upon maturation

<figure><img src="image_91.png" /></figure>

## iii) Pharma

Although genetics plays an important role in the
organisation of the different stages of wave
development (stages I, II, III; degradation of nicotinic
receptors, ...), dynamics is fundamental because it
orchestrates the structuring of the retina, through a
sequence of causally linked spatio-temporal events,
obeying simple principles and depending on a few
parameters.

<figure><img src="image_92.png" /></figure>

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

P4

P8
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);

* Spontaneous bursting occurs with a **probability very sensitive** to the position in region D.

* Waves propagate in a **random landscape** with **refractory zones** left by the previous waves, similar to forest fires although more complex, due to **the nonlinear coupling** between bursting cells.

* As a consequence, depending on the **maximal Ach conductance**, stage II retinal waves exhibit **different regimes of propagation** (anomalous diffusion, ballistic). The **distribution of sizes and durations** varies too, with a power law regime.

*Retinal waves shape the early visual system. Could they be used to train AI algorithms ?*

L. Pandley, S. Wood, J. Wood, Transformers self-organize like newborn visual systems when trained in prenatal worlds, arXiv:2601.03117v1

<figure><img src="image_94.png" /></figure>
How does lateral inhibition shape the
response to visual stimuli ?
Receptive Field

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

<figure><img src="image_96.png" /></figure>
Receptive Field

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

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

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

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

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

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

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

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

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

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

Source: https://www.pinterest.com/

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

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

Source: https://project.inria.fr/keops/
Receptive Field

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

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

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

Source: https://www.pinterest.com/

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

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

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

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

Source: https://project.inria.fr/keops/
# Receptive Field

Beyond the receptive field paradigm how does **lateral inhibition** and especially **amacrine cells** shape the collective response of the retinal network to spatio-temporal stimuli ?

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

<figure><img src="image_117.png" /></figure>

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

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

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

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

Light on

<figure><img src="image_122.png" /></figure>

<figure><img src="image_123.png" /></figure>

Light on

<figure><img src="image_124.png" /></figure>

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

Central illumination

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

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

Surround illumination

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

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

Diffuse illumination

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

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

1.5
How does lateral inhibition shape the
response to visual stimuli ?

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

<figure><img src="image_132.png" /></figure>
How does lateral inhibition shape the
response to visual stimuli ?

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

<figure><img src="image_133.png" /></figure>
How does lateral inhibition shape the response to visual stimuli ?

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

<figure><img src="image_134.png" /></figure>
How does lateral inhibition shape the response to visual stimuli ?

<figure><img src="image_135.png" /><figcaption>**Figure 3.1.** Designer Receptor Exclusively Activated by Designer Drugs. Left. Neurotransmitters bind to G protein-coupled receptors (GPCRs) of the membrane, which leads to a change in the activity of the neuron. Right. An engineered version of GPCRs, called DREADDs, responds only to a specific biologically inert chemical (a designer drug), hence allowing us to control neuronal activity. Source: https://benchling.com/pub/dreadds</figcaption></figure>
How does lateral inhibition shape the response to visual stimuli ?

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

DREADDs can be excitatory (hM3Dq - release of calcium from organelles leading to increase of intracellular concentration of free calcium, leading to membrane depolarisation and higher neuronal excitability), or inhibitory (hM4Di - opening of potassium channels, leading to membrane hyperpolarisation and neuronal silencing).
# How does lateral inhibition shape the response to visual stimuli ?

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

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

*What happens if one enhances the lateral inhibition with drugs?*

*Clozapine N Oxide (CNO) increases the activation level of amacrine cells in genetically modified mice*

*What are the consequences on the RGCs RF (STA)?*
# How does lateral inhibition shape the response to visual stimuli ?

*One experimentally observes that CNO affects the temporal STA of RGCs with, sometimes, drastic effects.*

## Small change in polarity

<figure><img src="image_138.png" /><figcaption>Control condition</figcaption></figure>

<figure><img src="image_139.png" /><figcaption>CNO</figcaption></figure>
# How does lateral inhibition shape the response to visual stimuli ?

*One experimentally observes that CNO affects the temporal STA of RGCs with, sometimes, drastic effects.*

## Big change in polarity

<figure><img src="image_140.png" /><figcaption>Control condition</figcaption></figure>

<figure><img src="image_141.png" /><figcaption>CNO</figcaption></figure>
# How does lateral inhibition shape the response to visual stimuli ?

*One experimentally observes that CNO affects the temporal STA of RGCs with, sometimes, drastic effects.*

*Big change in polarity*

<figure><img src="image_142.png" /><figcaption>Control condition</figcaption></figure>

<figure><img src="image_143.png" /><figcaption>CNO</figcaption></figure>
# How does lateral inhibition shape the response to visual stimuli ?

*One experimentally observes that CNO affects the temporal STA of RGCs with, sometimes, drastic effects.*

## Dissapearance of oscillations

<figure><img src="image_144.png" /><figcaption>Control condition</figcaption></figure>

<figure><img src="image_145.png" /><figcaption>CNO</figcaption></figure>
How does lateral inhibition shape the
response to visual stimuli ?

Can we explain these effects in a model ?

Working hypothesis : CNO modifies the **conductance** of specific ion channels and thereby the **characteristic time of ACs** as well as their **synaptic influence** to BCs and RGCs.
How does lateral inhibition shape the
response to visual stimuli ?

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

<figure><img src="image_146.png" /></figure>
How does lateral inhibition shape the
response to visual stimuli ?

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

<figure><img src="image_147.png" /></figure>
# How does lateral inhibition shape the response to visual stimuli ?

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

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

$$ 
\left\{ 
\begin{aligned}
\frac{dV_{B_i}}{dt} &= -\frac{1}{\tau_B}V_{B_i} - w^{-} \sum_{j=1}^{N_A} \Gamma_{B_i}^{A_j} V_{A_j} + \mathcal{P}_i(t), \quad i=1...N, \\
\frac{dV_{A_j}}{dt} &= -\frac{1}{\tau_A}V_{A_j} + w^{+} \sum_{i=1}^{N_B} \Gamma_{A_j}^{B_i} \mathcal{R}(V_{B_i}), \quad j=1...N, \\
\frac{dV_{G_k}}{dt} &= -\frac{1}{\tau_G}V_{G_k} + w_G^B \sum_{i=1}^{N_B} \Gamma_{G_k}^{B_i} \mathcal{R}(V_{B_i}) + w_G^A \sum_{j=1}^{N_A} \Gamma_{G_k}^{A_j} V_{A_j}, \quad k=1...N.
\end{aligned}
\right. 
$$
# How does lateral inhibition shape the response to visual stimuli ?

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

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

Stimulus dependent drive

$$ 
\left\{ 
\begin{aligned}
\frac{dV_{B_i}}{dt} &= -\frac{1}{\tau_B}V_{B_i} - w^{-} \sum_{j=1}^{N_A} \Gamma_{B_i}^{A_j} V_{A_j} + \mathcal{P}_i(t), && j=1 \dots N, \\
\frac{dV_{A_j}}{dt} &= -\frac{1}{\tau_A}V_{A_j} + w^{+} \sum_{i=1}^{N_B} \Gamma_{A_j}^{B_i} \mathcal{R}(V_{B_i}), && j=1 \dots N, \\
\frac{dV_{G_k}}{dt} &= -\frac{1}{\tau_G}V_{G_k} + w_G^B \sum_{i=1}^{N_B} \Gamma_{G_k}^{B_i} \mathcal{R}(V_{B_i}) + w_G^A \sum_{j=1}^{N_A} \Gamma_{G_k}^{A_j} V_{A_j}, && k=1 \dots N.
\end{aligned}
\right. 
$$
# How does lateral inhibition shape the response to visual stimuli ?

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

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

$$ 
\left\{ 
\begin{aligned}
\frac{dV_{B_i}}{dt} &= -\frac{1}{\tau_B} V_{B_i} - w^{-} \sum_{j=1}^{N_A} \Gamma_{B_i}^{A_j} V_{A_j} + \mathcal{P}_i(t), \quad i=1...N, \\
\frac{dV_{A_j}}{dt} &= -\frac{1}{\tau_A} V_{A_j} + w^{+} \sum_{i=1}^{N_B} \Gamma_{A_j}^{B_i} \mathcal{R}(V_{B_i}), \quad j=1...N, \\
\frac{dV_{G_k}}{dt} &= -\frac{1}{\tau_G} V_{G_k} + w_G^B \sum_{i=1}^{N_B} \Gamma_{G_k}^{A_i} \mathcal{R}(V_{B_i}), \quad w_G^A \sum_{j=1}^{N_A} \Gamma_{G_k}^{A_j} V_{A_j}, \quad k=1...N.
\end{aligned}
\right. 
$$

Non linearities:
gain control, rectification
# How does lateral inhibition shape the response to visual stimuli ?

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

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

## Inhibition

$$ \frac{dV_{B_i}}{dt} = -\frac{1}{\tau_B} V_{B_i} - w^- \sum_{j=1}^{N_A} \Gamma_{B_i}^{A_j} V_{A_j} \quad \mathcal{P}_i(t), \quad i=1...N, $$

$$ \frac{dV_{A_j}}{dt} = -\frac{1}{\tau_A} V_{A_j} + w^+ \sum_{i=1}^{N_B} \Gamma_{A_j}^{B_i} V_{B_i}, \quad j=1...N, $$

$$ \frac{dV_{G_k}}{dt} = -\frac{1}{\tau_G} V_{G_k} + w_G^B \sum_{i=1}^{N_B} \Gamma_{G_k}^{B_i} V_{B_i} - w_G^A \sum_{j=1}^{N_A} \Gamma_{G_k}^{A_j} V_{A_j}, \quad k=1...N. $$
# How does lateral inhibition shape the response to visual stimuli ?

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

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

$$ 
\left\{ 
\begin{aligned}
\frac{dV_{B_i}}{dt} &= -\frac{1}{\tau_B}V_{B_i} - w^{-} \sum_{j=1}^{N_A} \Gamma_{B_i}^{A_j} V_{A_j} + \mathcal{P}_i(t), \quad i=1...N, \\
\frac{dV_{A_j}}{dt} &= -\frac{1}{\tau_A}V_{A_j} + w^{+} \sum_{i=1}^{N_B} \Gamma_{A_j}^{B_i} V_{B_i}, \quad j=1...N, \\
\frac{dV_{G_k}}{dt} &= -\frac{1}{\tau_G}V_{G_k} + w_G^B \sum_{i=1}^{N_B} \Gamma_{G_k}^{B_i} V_{B_i} + w_G^A \sum_{j=1}^{N_A} \Gamma_{G_k}^{A_j} V_{A_j}, \quad k=1...N.
\end{aligned}
\right. 
$$

Excitation
# How does lateral inhibition shape the response to visual stimuli ?

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

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

Feedback inhibition

$$
\left\{
\begin{aligned}
\frac{dV_{B_i}}{dt} &= -\frac{1}{\tau_B} V_B - w^- \sum_{j=1}^{N_A} \Gamma_{B_i}^{A_j} V_{A_j} + \mathcal{P}_i(t), \quad i=1...N, \\
\frac{dV_{A_j}}{dt} &= -\frac{1}{\tau_A} V_{A_j} + w^+ \sum_{i=1}^{N_B} \Gamma_{A_j}^{B_i} V_{B_i}, \quad j=1...N, \\
\frac{dV_{G_k}}{dt} &= -\frac{1}{\tau_G} V_{G_k} + w_G^B \sum_{i=1}^{N_B} \Gamma_{G_k}^{B_i} V_{B_i} + w_G^A \sum_{j=1}^{N_A} \Gamma_{G_k}^{A_j} V_{A_j}, \quad k=1...N.
\end{aligned}
\right.
$$
# Neurotransmitter Receptors in the Retina

The retina, a light-detecting structure, occupies approximately 75% of the inner surface of the eye. Approximately 20-30 mm in diameter, the retina represents the only component of the central nervous system that can be observed without intervention. In humans, the retina is 0.5 mm in depth and composed of nine to ten definitive layers. These layers contain at least seven general cell types, including retinal pigment epithelial cells, photoreceptor, bipolar cells, horizontal cells, ganglion cells, Müller (gmu) cells, and amacrine cells. Multiple subtypes also exist for each of these cell types. To date, there are at least 10 distinct bipolar cells, between 10 and 15 ganglion cells, and between 25 and 30 different amacrine cells. Additionally, two types of photoreceptors, rod and cone cells, are found in the retina. Rod cells account for about 95% of all photoreceptors and are concentrated at the outer edges of the retina while cone cells are concentrated near the center of the retina around an area called the macula. Rod and cone cells contain a light-sensitive pigment called Rhodopsin and Phosphoinositide, respectively. These photoprotections are comprised of Opabin, a seven transmembrane G protein-coupled receptor, covalently bound to a Vitamin A derivative termed Retinal, and absorb specific wavelengths of light. Rhodopsin is sensitive to blue-green light while Photopsins, depending

on their exact structure, will absorb red, green, or blue light. Rod cells are highly sensitive photoreceptors and are associated with scotopic (night) vision. Cone cells, which contain only one of the three possible Photopsin molecules, are less sensitive to light and are responsible for color vision. An additional photopigment, known as Melanopsin or Opabin, has been detected in specialized ganglion cells. This photopigment detects light in the blue spectrum and reveals information to the non-image forming part of the visual system.
Phototransduction begins when light falls on photoreceptors and activates the photopigment. This initiates a signal transduction cascade in photoreceptors, which is then transmitted to horizontal cells and bipolar cells. Bipolar cells subsequently interact with ganglion cells and amacrine cells, and the resulting processed information leaves the retina via ganglion cell axons, which merge to form the optic nerve. This schematic depicts potential neural processes, their interactions, and associated neurotransmitter receptors in the retina. R&D Systems currently offers antibodies to many of these molecules.

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

<figure><img src="image_155.png" /></figure>

KEY:

<table><tr><td>Adenosine A2A Receptor (AdORA2A)</td><td>GABA-B Receptor</td><td>Metabotropic Glutamate Receptor (mGluR)</td><td>Optic Receptor</td></tr><tr><td>AMPA Receptor (SCh)</td><td>Glutamate Receptor, delta (GluRδ)</td><td>Muscarinic Acetylcholine Receptor (mAChR)</td><td>Oxin Receptor 1</td></tr><tr><td>Calcium Receptor Receptor (CRhL)</td><td>Glutamate Receptor (GluR)</td><td>Neurotransmitter 1 Receptor (ntrL1)</td><td>Oxin Receptor 2 (OX2 and P2Y)</td></tr><tr><td>Dopamine Receptor (DR)</td><td>Histamine Receptor (HHR)</td><td>Neurotransmitter Y Receptor (nYR)</td><td>Somatostatin Receptor (SSTR)</td></tr><tr><td>Erythropoietin Receptor (EpoR)</td><td>Kainate Receptor (GluRδ/Kainin and KA/GRIK)</td><td>Neurotransmitter Acetylcholine Receptor (nAChR)</td><td>Tri Receptor Kinase (TriK)</td></tr><tr><td>GABA-A Receptor</td><td>Metabotropic Glutamate Receptor 1 (mGluR1)</td><td>NMDA Receptor (Ntr)</td><td></td></tr></table>
# Neurotransmitter Receptors in the Retina

The retina, a light-detecting structure, occupies approximately 75% of the inner surface of the eye. Approximately 20-30 mm in diameter, the retina represents the only component of the central nervous system that can be observed without intervention. In humans, the retina is 0.5 mm in depth and composed of nine to ten definitive layers. These layers contain at least seven general cell types, including retinal pigment epithelial cells, photoreceptors, bipolar cells, horizontal cells, ganglion cells, Müller (glia) cells, and amacrine cells. Multiple subtypes also exist for each of these cell types. To date, there are at least 10 distinct bipolar cells, between 10 and 15 ganglion cells, and between 25 and 30 different amacrine cells. Additionally, two types of photoreceptors, rod and cone cells, are found in the retina. Rod cells account for about 95% of all photoreceptors and are concentrated at the outer edges of the retina while cone cells are concentrated near the center of the retina around an area called the macula. Rod and cone cells contain a light-sensitive pigment called Rhodopsin and Photopsin, respectively. These photopigments are comprised of Opn, a seven transmembrane β protein-coupled receptor, covalently bound to a Vitamin A derivative termed Retinal, and absorb specific wavelengths of light. Rhodopsin is sensitive to blue-green light while Photopsins, depending

on their exact structure, will absorb red, green, or blue light. Rod cells are highly sensitive photoreceptors and are designated (right) vision. Cone cells, which contain only one of the three possible Photopsin molecules, are less sensitive to light and are responsible for color vision. An additional photopigment, known as Melanopsin or Opn4, has been detected in specialized ganglion cells. This photopigment detects light in the blue spectrum and relays information to the non-image forming part of the visual system.
Phototransduction begins when light falls on photoreceptors and activates the photopigment. This initiates a signal transduction cascade in photoreceptors, which is then transmitted to horizontal cells and bipolar cells. Bipolar cells subsequently interact with ganglion cells and amacrine cells, and the resulting processed information leaves the retina via ganglion cell axons, which merge to form the optic nerve. This schematic depicts potential neuronal processes, their interactions, and associated neurotransmitter receptors in the retina. R&D Systems currently offers antibodies to many of these molecules.

<figure><img src="image_156.png" /></figure>

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

Here, we do not want to reproduce a specific retina circuit with feedback.

Instead, we want to infer the consequences of a potential feedback mechanism between excitation and inhibition.

The feedback intensity is controlled by a unique dimensionless parameter.

$$ \eta = w^{-} w^{+} \tau_{A} \tau_{B}. $$
# How does lateral inhibition shape the response to visual stimuli ?

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

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

Feedback inhibition

$$ \frac{dV_{B_i}}{dt} = \left( -\frac{1}{\tau_B} V_{B_i} + w^- \sum_{j=1}^{N_A} \Gamma_{B_i}^{A_j} V_{A_j} + \mathcal{P}_i(t), \quad i=1...N, \right) $$

$$ \frac{dV_{A_j}}{dt} = \left( -\frac{1}{\tau_A} V_{A_j} + w^+ \sum_{i=1}^{N_B} \Gamma_{A_j}^{B_i} V_{B_i}, \quad j=1...N, \right) $$

$$ \frac{dV_{G_k}}{dt} = -\frac{1}{\tau_G} V_{G_k} + w_G^B \sum_{i=1}^{N_B} \Gamma_{G_k}^{B_i} V_{B_i} + w_G^A \sum_{j=1}^{N_A} \Gamma_{G_k}^{A_j} V_{A_j}, \quad k=1...N. $$

$$ \eta = w^- w^+ \tau_A \tau_B. $$
# How does lateral inhibition shape the response to visual stimuli ?

Can we explain these effects in a model ?

**Working hypothesis** : CNO modifies the **conductance** of specific ion channels and thereby the **characteristic time of ACs** as well as their **synaptic influence** to BCs and RGCs.
# How does lateral inhibition shape the response to visual stimuli ?

*One experimentally observes that CNO affects the temporal STA of RGCs with, sometimes, drastic effects.*

*Small change in polarity*

<figure><img src="image_159.png" /><figcaption>Control condition</figcaption></figure>

<figure><img src="image_160.png" /><figcaption>CNO</figcaption></figure>
# How does lateral inhibition shape the response to visual stimuli ?

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

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

<figure><img src="image_163.png" /><figcaption>Control condition</figcaption></figure>

<figure><img src="image_164.png" /><figcaption>CNO</figcaption></figure>
# How does lateral inhibition shape the response to visual stimuli ?

*One experimentally observes that CNO affects the temporal STA of RGCs with, sometimes, drastic effects.*

## Big change in polarity

<figure><img src="image_165.png" /><figcaption>Control condition</figcaption></figure>

<figure><img src="image_166.png" /><figcaption>CNO</figcaption></figure>
# How does lateral inhibition shape the response to visual stimuli ?

<figure><img src="image_167.png" /></figure>

<figure><img src="image_168.png" /></figure>

<figure><img src="image_169.png" /></figure>

<figure><img src="image_170.png" /></figure>

Control condition

CNO
# How does lateral inhibition shape the response to visual stimuli ?

*One experimentally observes that CNO affects the temporal STA of RGCs with, sometimes, drastic effects.*

## Dissapearance of oscillations

<figure><img src="image_171.png" /><figcaption>Control condition</figcaption></figure>

<figure><img src="image_172.png" /><figcaption>CNO</figcaption></figure>
# How does lateral inhibition shape the response to visual stimuli ?

<figure><img src="image_173.png" /></figure>

<figure><img src="image_174.png" /></figure>

<figure><img src="image_175.png" /></figure>

<figure><img src="image_176.png" /></figure>

Control condition

CNO
# How does lateral inhibition shape the response to visual stimuli ?

<figure><img src="image_177.png" /></figure>

A

<figure><img src="image_178.png" /></figure>

<figure><img src="image_179.png" /></figure>

<figure><img src="image_180.png" /></figure>

<figure><img src="image_181.png" /></figure>

B

<figure><img src="image_182.png" /></figure>

<figure><img src="image_183.png" /></figure>

D

<figure><img src="image_184.png" /></figure>

<figure><img src="image_185.png" /></figure>

<figure><img src="image_186.png" /></figure>

C
How does lateral inhibition shape the
response to visual stimuli ?

<figure><img src="image_187.png" /></figure>

Control condition
How does lateral inhibition shape the
response to visual stimuli ?

<figure><img src="image_188.png" /></figure>

Application of CNO (Clozapine N Oxide)
# How does lateral inhibition shape the response to visual stimuli ?

<figure><img src="image_189.png" /></figure>

<figure><img src="image_190.png" /></figure>

Control condition

<figure><img src="image_191.png" /></figure>

<figure><img src="image_192.png" /></figure>

<figure><img src="image_193.png" /></figure>
# How does lateral inhibition shape the response to visual stimuli ?

<figure><img src="image_194.png" /></figure>

<figure><img src="image_195.png" /></figure>

Control condition

<figure><img src="image_196.png" /></figure>

CNO

<figure><img src="image_197.png" /></figure>

<figure><img src="image_198.png" /></figure>
How does lateral inhibition shape the
response to *moving* stimuli ?
# How does lateral inhibition shape the response to moving stimuli ?

<figure><img src="image_199.png" /></figure>
How does lateral inhibition shape the response to moving stimuli ?

<figure><img src="image_200.png" /></figure>
How does lateral inhibition shape the response to moving stimuli ?

<figure><img src="image_201.png" /></figure>
How does lateral inhibition shape the response to moving stimuli ?

<figure><img src="image_202.png" /></figure>
How does lateral inhibition shape the response to moving stimuli ?

<figure><img src="image_203.png" /></figure>
# How does lateral inhibition shape the response to moving stimuli ?

<figure><img src="image_204.png" /></figure>

## Conjecture :

A moving object triggers **waves of activity** in the retina (and in the visual cortex).

These waves are shaped and modulated by **lateral inhibition**.

The conjunction of the moving stimulus and of the forward/backward and excitatory/inhibitory effects **selects** spatial wave lengths and temporal frequencies leading to **specific responses** and **resonance effects**.
How does lateral inhibition shape the response to moving stimuli ?

<figure><img src="image_205.png" /></figure>
Visual Anticipation

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

<figure><img src="image_206.png" /></figure>

« Anticipatory responses along motion trajectories in awake 2 monkey area V1”, Benvenutti et al. 2020, submitted to Elife and https://www.biorxiv.org/content/10.1101/2020.03.26.010017v2.full.pdf
Visual Anticipation

**Anticipation** also takes place in the retina

<figure><img src="image_207.png" /></figure>

Source: M. Berry, I. Brivanlou, T. Jordan, and M. Meister. Anticipation of moving stimuli by the retina. Nature, 398(6725):334–338, 1999
Lateral inhibition can induce retinal
anticipation

S. Ebert, B. Cessac, *Distinct inhibitory connectivity motifs trigger distinct forms of anticipation in the retinal network*, submitted.

<figure><img src="image_208.png" /></figure>

<figure><img src="image_209.png" /></figure>

<figure><img src="image_210.png" /></figure>

<figure><img src="image_211.png" /></figure>

<figure><img src="image_212.png" /></figure>

<figure><img src="image_213.png" /></figure>

<figure><img src="image_214.png" /></figure>
Lateral inhibition can induce retinal
anticipation

S. Ebert, B. Cessac, Distinct inhibito-
submitted.

Impulse responses are very
different in the feedforward
and the feedback case.

anticipation in the retinal network,

A

<figure><img src="image_215.png" /></figure>

B

<figure><img src="image_216.png" /></figure>

E

<figure><img src="image_217.png" /></figure>

C

<figure><img src="image_218.png" /></figure>

D

<figure><img src="image_219.png" /></figure>

F

<figure><img src="image_220.png" /></figure>

G

<figure><img src="image_221.png" /></figure>
Lateral inhibition can induce retinal
anticipation

S. Ebert, B. Cessac, Distinct inhibitory connectivity motifs trigger distinct forms of anticipation in the retinal network, submitted.

<figure><img src="image_222.png" /></figure>

<figure><img src="image_223.png" /></figure>

<figure><img src="image_224.png" /></figure>
Lateral inhibition can induce retinal
anticipation

S. Ebert, B. Cessac, Distinct inhibito-
submitted.

Responses to moving bar
differ too.

anticipation in the retinal network,

A

stimulus

t<sub>bar</sub>

forward

<figure><img src="image_225.png" /></figure>

recurrent

space [mm]

<figure><img src="image_226.png" /></figure>

B

<figure><img src="image_227.png" /></figure>

C

<figure><img src="image_228.png" /></figure>
Lateral inhibition can induce retinal
anticipation

S. Ebert, B. Cessac, Distinct inhibito-
submitted.

Responses to moving bar
differ too.

anticipation in the retinal network,

A

stimulus

tbar

forward

<figure><img src="image_229.png" /></figure>

recurrent

<figure><img src="image_230.png" /></figure>

<figure><img src="image_231.png" /></figure>

Appearance of oscillations.

B

<figure><img src="image_232.png" /></figure>

C

<figure><img src="image_233.png" /></figure>
# Lateral inhibition can induce retinal anticipation

S. Ebert, B. Cessac, Distinct inhibition in the retinal network, submitted.

Responses to moving bar differ too.

A

<figure><img src="image_234.png" /></figure>

B

<figure><img src="image_235.png" /></figure>

recurrent

<figure><img src="image_236.png" /></figure>

Appearance of oscillations.

C

<figure><img src="image_237.png" /></figure>

Preferred speed
Lateral inhibition can induce retinal
anticipation

S. Ebert, B. Cessac, Distinct inhibitory connectivity motifs trigger distinct forms of anticipation in the retinal network, submitted.

<figure><img src="image_238.png" /></figure>

<figure><img src="image_239.png" /></figure>

<figure><img src="image_240.png" /></figure>

<figure><img src="image_241.png" /></figure>

<figure><img src="image_242.png" /></figure>

<figure><img src="image_243.png" /></figure>
Lateral inhibition can induce retinal
anticipation

S. Ebert, B. Cessac, Distinct inhibitory
submitted.

Response profile and
preferred speed depend
on the inhibitory strength.

of anticipation in the retinal network,

<figure><img src="image_244.png" /></figure>

<figure><img src="image_245.png" /></figure>

<figure><img src="image_246.png" /></figure>

<figure><img src="image_247.png" /></figure>

<figure><img src="image_248.png" /></figure>

<figure><img src="image_249.png" /></figure>
The spatio-temporal response pattern
depends on the stimulus speed

<figure><img src="image_250.png" /></figure>
The spatio-temporal response pattern
depends on the stimulus speed

<figure><img src="image_251.png" /></figure>
The spatio-temporal response pattern
depends on the stimulus speed

<figure><img src="image_252.png" /></figure>
The spatio-temporal response pattern
depends on the stimulus speed

<figure><img src="image_253.png" /></figure>
# The spatio-temporal response pattern depends on the stimulus speed

Oscillations correspond to interference between waves and resonances.

<figure><img src="image_254.png" /></figure>

<figure><img src="image_255.png" /></figure>
# How does lateral inhibition shape the response to moving stimuli ?

<figure><img src="image_256.png" /></figure>

## Conjecture :

A moving object triggers **waves of activity** in the retina (and in the visual cortex).

These waves are shaped and modulated by **lateral inhibition**.

The conjunction of the moving stimulus and of the forward/backward and excitatory/inhibitory effects **selects** spatial wave lengths and temporal frequencies leading to **specific responses** and **resonance effects**.
Conclusions
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 **lateral connectivity** and especially **Amacrine cells** contribute to the generation of **spatio-temporal patterns (waves)**, tuned by a few **parameters**.
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 **lateral connectivity** and especially **Amacrine cells** contribute to the generation of **spatio-temporal patterns (waves)**, tuned by a few **parameters**.

* During **development** these waves orchestrate the retinal network structuration.
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 **lateral connectivity** and especially **Amacrine cells** contribute to the generation of **spatio-temporal patterns (waves)**, tuned by a few **parameters**.

* During **development** these waves orchestrate the retinal network structuration.

* In adult retina, waves triggered by **moving objects** induce prominent mechanisms such as **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 **lateral connectivity** and especially **Amacrine cells** contribute to the generation of **spatio-temporal patterns (waves)**, tuned by a few **parameters**.

* During **development** these waves orchestrate the retinal network structuration.

* In adult retina, waves triggered by **moving objects** induce prominent mechanisms such as **anticipation**.

* Waves induce **spatio-temporal correlations** that may contribute to better **encoding** a visual scene, with **low energy consumption**.
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 **lateral connectivity** and especially **Amacrine cells** contribute to the generation of **spatio-temporal patterns (waves)**, tuned by a few **parameters**.

* During **development** these waves orchestrate the retinal network structuration.

* In adult retina, waves triggered by **moving objects** induce prominent mechanisms such as **anticipation**.

* Waves induce **spatio-temporal correlations** that may contribute to better **encoding** a visual scene, with **low energy consumption**.

Waves could be the fundamental key to understand the **visual processing** and **propose new algorithms**, with **low energy consumption**, to improve **artificial vision**.