A mathematical approach to retinal waves

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A biophysical model explains the spontaneous bursting behavior in the developing retina

During early development, waves of activity propagate across the retina and play a key role in the proper wiring of the early visual system. During the stage II these waves are triggered by a transient network of neurons, called Starburst Amacrine Cells (SACs), showing a bursting activity which disappears upon further maturation. While several models have attempted to reproduce retinal waves, none of them is able to mimic the rhythmic autonomous bursting of individual SACs and reveal how these cells change their intrinsic properties during development. Here, we introduce a mathematical model, grounded on biophysics, which enables us to reproduce the bursting activity of SACs and to propose a plausible, generic and robust, mechanism that generates it. The core parameters controlling repetitive firing are fast depolarizing $V$-gated calcium channels and hyperpolarizing $V$-gated potassium channels. The quiescent phase of bursting is controlled by a slow after hyperpolarization (sAHP), mediated by calcium-dependent potassium channels. Based on a bifurcation analysis we show how biophysical parameters, regulating calcium and potassium activity, control the spontaneously occurring fast oscillatory activity followed by long refractory periods in individual SACs. We make a testable experimental prediction on the role of voltage-dependent potassium channels on the excitability properties of SACs and on the evolution of this excitability along development. We also propose an explanation on how SACs can exhibit a large variability in their bursting periods, as observed experimentally within a SACs network as well as across different species, yet based on a simple, unique, mechanism. As we discuss, these observations at the cellular level have a deep impact on the retinal waves description.Comment: 25 pages, 13 figures, submitte

Biophysical reaction-diffusion model for stage II retinal waves and bifurcations analysis

International audienceRetinal waves are spontaneous waves of spiking activity observed in the retina, during development only,playing a central role in shaping the visual system and retinal circuitry. Understanding how these waves areinitiated and propagate in the retina could enable one to control, guide and predict them in the in vivo adultretina as inducing them is expected to reintroduce some plasticity in the retinal tissue and in the projectionsto the LGN. In this context, we propose a physiologically realistic reaction-diffusion model for the mechanismsof the emergence of stage II cholinergic retinal waves during development. We perform the bifurcation analysiswhen varying two biophysically relevant parameters, the conductances of calcium and potassium g_Ca,g_K respectively. The two main goals of our work are: firstly, reproduce the experimental recordings ofdevelopmental retinal waves by simulating our model and secondly, explore the different dynamical behavioursobserved when varying these two parameters

Modelling spontaneous propagating waves in the early retina

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Modeling the emergence of stage II retinal waves in immature retina

International audienceRetinal waves are spontaneous bursting activity propagating in the developping retina until vision is functional. In this work we propose a biophysical modelling of the mechanism that generates the spontaneous intrinsic cell-autonomous rhythmic bursting in Starbust Amacrine Cells (SACs), observed experimentally in [1] which is directly linked with the emergence of stage II retinal waves. We analyze this system from the dynamical system and bifurcation theory perspective

Biophysical modelling of the intrinsic mechanisms of the autonomous starbust cells during stage II retinal waves

International audienceRetinal waves are spontaneous bursting activity propagating in the developping retina until vision is functional. In this work we propose a biophysical modelling of the mechanism that generates the spontaneous intrinsic cell-autonomous rhythmic bursting in Starbust Amacrine Cells (SACs), observed experimentally in [1] which is directly linked with the emergence of stage II retinal waves. We analyze this system from the dynamical system and bifurcation theory perspective

Pattern formation and criticality in the developing retina

International audienceIn the early retina, spontaneous collective network activity emerges as propagating waves, playing a central role in shaping the visual system. Elucidating how the characteristics of such waves depend on biophysical parameters, would help us understand the underlying mechanisms of spatio-temporal patterns formation in the developing retina and their role in shaping the visual system. We have elaborated a set of detailed biophysical equations for a network of retinal cells coupled with excita-tory lateral cholinergic connections, close enough to reality to reproduce and predict experimental results. From bifurcation theory, we predict that there exists a regime of parameters for which the network of cells in the developing retina is a critical system. This property is manifested via power law distributions for the waves characteristics (i.e. waves size), meaning that waves statistics could exhibit maximal variability. This critical regime is analytically characterized, predicting the exact form of the critical coupling strength of cells. Away from this regime of parameters, no power-law like distributions are observed. This theoretical result is in agreement with our experimental recordings in perinatal mice, revealing power-laws as well, suggesting that there exists a mechanism setting the retinal cells close to this critical regime. Context & Motivation Retinal waves characteristics exhibit a vast variability: Questions: 1. How do the variable characteristics of retinal waves depend on few biophysical parameters? 2. How can we characterize quantitatively the different dynamical regimes and the transitions between them? 3. Why is it important for the early retinal network to exhibit large variability in the characteristics of spatiotemporal patterns? 4. What are the biophysical mechanisms of the spatiotemporal patterns formation in the early retina? Patterns vary upon parameters variation Having proposed a biophysical model for retinal waves [1], we use our equations to understand the underlying mechanisms of waves apparition and propagation: Analytic condition for wave propagation Waves propagation analytic condition for a critical threshold of cholinergic coupling Based on bifurcation theory: 1. We derive analytic forms for a critical waves propagation threshold of coupling strength among cells g A C , and for the waves speed (not shown). 2. We propose a possible mechanism of how power-law distributions could appear near this propagation threshold, where the cell is in fact close to a (saddle-node) bifurcation point. At this point, dynamics are driven mainly by noise fluctuations, leading to maximum variance in the patterns characteristics (e.g. waves size), manifesting power-law like distributions, and therefore indicating possible links to criticality. Finding power-laws in experiments We performed MEA (256 electrodes) experiments on P5 mice (stage II retinal waves) at Vision Institute, Paris. • A power-law distribution for the waves size is computed at the regime where the transition occurs in our model (B), matching our experimental data on P5 mice. • This indicates that maybe the network of SACs is naturally set close to a critical state by a possible homeostatic mechanism, yet to be identified. Conclusions and Perspectives • Our model allows us to anticipate how biophysical parameters variations (e.g. conductance) may impact the characteristics of waves. • We predict that SACs are close to a bifurcation point, leading to explaining the different types of variability of retinal waves as well as proposing encouraging, although still primary links to criticality. • Further analysis is needed to characterize critical systems, such as studying in detail possible phase transitions, and computing critical exponents on the theoretical side. • On the experimental side, new and more precise methods should be proposed for the exact characterization of power-law distributions in experimental recordings. • Extend our phenomenological model in order to identify the possible homestatic mechanism that drives the network to a critical state. • Explore the role of the indicated criticality in the early retina, possibly related to an optimizing the response sensitivity to multi-scale stimuli upon matura-tion, enhancing the dynamical range of the early network (Steven's law)

Classifying the spatiotemporal patterns within stage II retinal waves through dynamical systems analysis

International audienceRetinal waves are bursts of activity occurring spontaneously in the developing retina of vertebrate species, contributing to the shaping of the visual system organization. They are characterized by localized groups of neurons becoming simultaneously active, initiated at random points. Based on our previous modelling work [1], we now propose a classification of stage II retinal waves patterns as a function of acetylcholine coupling strength and a possible mechanism for waves generation. Our model predicts that spatiotemporal patterns evolve upon maturation or pharmacological manipulation and that there is a regime of cholinergic coupling, only for which, waves are characterized by power-law distributions

Biovision project-team: Biological vision: integrative models and vision aid systems for visually impaired people

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