Cessac, Bruno

Gil, Lionel

Matzakos-Karvouniari, Dora

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HAL Id: hal-02268281https://hal.inria.fr/hal-02268281Submitted on 21 Aug 2019HAL is a multi-disciplinary open accessarchive for the deposit and dissemination of sci-entific research documents, whether they are pub-lished or not. The documents may come fromteaching and research institutions in France orabroad, or from public or private research centers.L’archive ouverte pluridisciplinaire HAL, estdestinée au dépôt et à la diffusion de documentsscientifiques de niveau recherche, publiés ou non,émanant des établissements d’enseignement et derecherche français ou étrangers, des laboratoirespublics ou privés.Modelling spontaneous propagating waves in the earlyretinaBruno Cessac, Dora Matzakos-Karvouniari, Lionel GilTo cite this version:Bruno Cessac, Dora Matzakos-Karvouniari, Lionel Gil. Modelling spontaneous propagating waves inthe early retina. Waves Côte d’azur, Jun 2019, Nice, France. ￿hal-02268281￿Dora Matzakos-Karvouniari, LJAD, UCA Lionel Gil, INPHYNI, UCA Bruno Cessac, Biovision, InriaModelling spontaneous propagating  waves in the early retina  2Visual systemLightLateral Geniculate  Nucleus (LGN)RetinaVisual  Cortex3The structure of the adult retinaLightphotoreceptorsganglion cells bipolar cellsamacrine cells horizontal cellsRetina’s layered structure4The structure of the retina during developmentLightXRetina’s layered structure is shaped during developmentBut How? Retinal waves!5What are Retinal waves?Recordings from the retinaMulti-electrode array(MEA)(Maccione et al. 2014)ΜΕΑ recording of the voltage from a P11 mouse retina in the presence of 10 μM bicuculline. Spontaneous spatio-temporal waves during development Disappear short  after birth when vision is functional6Stage INon SynapticAcetylcholine (nAchR)Glutamate• Formation of early retinal circuitry • Chemical synapses are not formed yet • Electrical synapses mediatedStage II• Retinotopic mapping • Eye specific segregation • Nicotinic Acetylcholine Receptors (nAchR)Stage III• Disappear when vision is functional • Glutamate/AMPA ReceptorsThe timeline of retinal wavesP0 P10 P14 *mouse timelineAge (postnatal days)Eye opening7Acetylcholine (nAchR)P0 P10 P14Age (postnatal days)Focusing on stage II retinal waves*mouse timeline8Why modelling retinal waves is an interesting problem?• Retinal waves instruct the shaping of the visual system• Understanding the mechanisms that generate them may help to control them pharmacologically.• Using biophysical modelling we could  i) find generic principles    ii) explain experiments and propose new ones  leading to the better understanding of the underlying mechanisms of waves.• The mathematical modelling of the dynamics of a complex biological process is a very interesting physical problem with real consequences in biologyVariability within retinal wavesZheng et al. 2004i) Across speciesRabbitMouseChickTurtleii) DevelopmentGodfrey et al. 20070150300450600Speed (μm/s)iii) Pharmacology iv) Spatial VariabilityP4P8Time (sec)Normal Isolated SAC bursts at P4Normal Isolated SAC no bursting at P8Pharmacology restores bursting500Zheng et al. 2006Maccione et al. 2014Waves have variable shapes due to a refractory mechanism which controls their borders. It is called sAHP (slow AfterHyper Polarization)  for stage II. 037,575112,5150Period (s)Modelling stage II retinal waves dynamics 101. D. Matzakos-Karvouniari et al., A biophysical model explains the bursting activity in the immature retina, Nat Sci Rep, 201911The biophysics of stage II retinal wavesNicotinic Acetylcholine ReceptorsMutual Excitatory synapses Spontaneous Fast oscillations sAHP (slow AfterHyperpolarization) *Calcium regulated*Individual SACs burst spontaneouslyImmature SACs form a transient cholinergic networkZheng et al. 2006 12A model for stage II retinal waves6 variables: V, N, C, R, S, A 3 time scales: Fast, Medium, Slow Nonlinear cholinergic coupling ~30 parameters  FastMediumSlowMatzakos-Karvouniari et al. 2019, Lansdell et al. 2014, Hennig et al. 2009, Morris-Lecar 198113V (mV)Cd2+Experiment (Zheng et al. 2006) ModelA BTEAValidating our equations by reproducing  experimentsI. Equations reproduce the role of voltage-gated Ca+2 channels on fast oscillations II. We predict that a fast potassium channel is needed to produce fast oscillationsAnalysis through bifurcation theory  How do waves start?141. D. Matzakos-Karvouniari et al., A biophysical model explains the bursting activity in the immature retina, Nat Sci Rep, 201915The mechanism of bursting revealed through bifurcationsFast subsystem16The mechanism of bursting revealed through bifurcationsFast subsystem17Limit cycles and Homoclinic OrbitIext (pA)NV (mV)1HopfSaddle - NodeHomoclinicNeutralSaddle*0-60-10250Bifurcation diagram of the fast subsystem (V,N)3DWhat do we learn about SACs?Stable rest state co existing with  unstable fixed pointsCell oscillates constantly due to  a limit cycleOnly stable rest state at a high voltageThe rest state of SACs is near a bifurcation point!I. The SACs repertoire of dynamics upon a current application18Limit cycles and Homoclinic OrbitIext (pA)NV (mV)1HopfSaddle - NodeHomoclinicNeutralSaddle*0-60-10250Bifurcation diagram of the fast subsystem (V,N)3DWhat do we learn about SACs?I. The SACs repertoire of dynamics upon a current applicationZoom in this  interesting region19Limit cycles and Homoclinic OrbitIext (pA)NV (mV)1HopfSaddle - NodeHomoclinicNeutralSaddle*0-60-10250Bifurcation diagram of the fast subsystem (V,N)3DWhat do we learn about SACs?I. The SACs repertoire of dynamics upon a current applicationZoom in this  interesting region20The mechanism of bursting revealed through bifurcationsLimit cycleStable fixed pointsWhat do we learn about SACs?II. The role of sAHP for the bursting mechanismBifurcation diagram of the fast subsystem V,NUnstable  fixed points21The mechanism of bursting revealed through bifurcationsLimit cycleStable fixed pointsUnstable  fixed pointsTrajectory of the full systemWhat do we learn about SACs?II. The role of sAHP for the bursting mechanismBifurcation diagram of the fast subsystem V,NDue to time scale separation, we make the approximation and superimpose the trajectory of the full system on the bifurcation diagram]22The mechanism of bursting revealed through bifurcationsThe biophysical interpretationHow do waves propagate and stop?232. D. Karvouniari et al., Spontaneous emergence of spatio-temporal structure in the early retina, (under preparation for Phys. Review E)24How do waves propagate?• Full interacting waves are a complex paradigm • Waves propagate in a landscape changing constantly • It is hard to extract a mechanism of propagation  25A network model of stage II retinal wavesCondition for wave propagation: IAch+IsAHP=ISNLimit cycleStable fixed pointsTrajectory of the full systemIAch+IAch+IsAHP27The coupled dynamics of 2 SACsCell #1Cell #2Ach profileCondition for bursting to start: IAch+IsAHP=ISNResponse TimeWaves speed28• For the zero order approximation we assumed that V_, R and Ω do not depend on gA. Fit holds for smaller gA• Waves speed (1D)gA(nS)v (cells)(critical point)Analytic form            0 order2nd orderSimulationBased on our model, where wave propagation is only possible when the total current received by a  neighbouring cell exceeds the (saddle-node)  bifurcation threshold, we extract the analytic form for the  waves propagation coupling threshold (1D case).Analytic form of waves propagation coupling threshold29Propagation in 1D without frictionFitting trajectory: u : speed z : propagation exponent (e.g. z=1 ballistic)• Around the propagation threshold, waves could stop even without friction, due to noise and the fact that the Ach current is not sufficient to fulfil the propagation condition30 Approximation, free propagation no friction, all cells are at restPropagation in 2D without friction• The speed increases with  the coupling strengthpropagation  threshold• Waves speed (2D)31How do waves stop? The sAHP sagaMax Waves size as a  function of the strength of sAHP+—SimulationAnalytic FitThe waves size decreases hyperbolically with the strength of sAHPExample of a wave propagation  in a radial landscape of R32 gA (nS)Patterns evolve within stage IIstage IIWeak Moderate StrongSynchronisation Transition Average population firing rateThe average and the std of the population firing rate undergo a transition at a specific range of cholinergic couplingStandard deviation firing rate<F>g_A g_AσSynchronisation Transition Our model indicates the existence of power law like distributions for waves sizes for a range of coupling strengthDistribution of waves sizes Fano coefficientRatio of mean and standard deviation of a distribution35Indications for criticality in experiments• Criticality is a precise concept in physics and power law distributions are not enough to characterise itHennig et al. 2009• Hennig et. al 2009 also found power law -like distributions for the waves sizes•       Also, criticality is not a natural state and it is achieved by bringing the system to a specific point Two “critical” points: i) Experimental data need to be analysed in a more elaborate way to reveal the actual distributions of waves characteristics ii) If indeed there is criticality and power laws, then we miss a mechanism that brings the system to the critical point36Conclusions & Perspectives• Analysing the dynamics of our model using dynamical systems theory leading to a main hypothesis: Immature SACs are near a bifurcation point!  • Consequences: • individual SACs properties* • how SACs lose their excitability upon maturation • explain variability of wave features across species • wave propagation conditions* • criticality and power laws* • spontaneous activity could be generated by a SACs network that is optimised for input processing=dynamic range • Future direction towards completing studying waves in a dynamically changing landscape • Characterise criticality in our model and use the same tools to characterise experimental data• Proposing new experiments based on theoretical hypothesisThank you all  :)37

Modelling spontaneous propagating waves in the early retina

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Modelling spontaneous propagating waves in the early retina

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