Noise driven broadening of the neural synchronisation transition in stage II retinal waves Dora Matzakos-Karvouniari1;3, Bruno Cessac1 and L. Gil2 1 Universit´eC^oted'Azur, Inria, Biovision team, France 2 Universit´eC^oted'Azur, Institut de Physique de Nice (INPHYNI), France 3 Universit´eC^oted'Azur, Laboratoire Jean-Alexandre Dieudonn´e(LJAD), France (Dated: December 10, 2019) Based on a biophysical model of retinal Starburst Amacrine Cell (SAC) [1] we analyse here the dynamics of retinal waves, arising during the visual system development. Waves are induced by spontaneous bursting of SACs and their coupling via acetycholine. We show that, despite the acetylcholine coupling intensity has been experimentally observed to change during development [2], SACs retinal waves can nevertheless stay in a regime with power law distributions, reminiscent of a critical regime. Thus, this regime occurs on a range of coupling parameters instead of a single point as in usual phase transitions. We explain this phenomenon thanks to a coherence-resonance mechanism, where noise is responsible for the broadening of the critical coupling strength range. PACS numbers: 42.55.Ah, 42.65.Sf, 32.60.+i,06.30.Ft I. INTRODUCTION This captivating idea raises nevertheless the following important issue. The spontaneous stage II retinal waves Since the seminal work of Beggs and Plen [3] - report- are mediated by a transient network of SACs, connected ing that neocortical activity in rat slices occur in the through excitatory cholinergic connections [35], which are form of neural avalanches with power law distributions formed only during a developmental window up to their close to a critical branching process - there have been complete disappearance. Especially, in [2], Zheng et al numerous papers suggesting that the brain as a dynam- have shown that the intensity of the acetylcholine cou- ical system fluctuates around a critical point. Scale-free pling is monotonously decreasing with time. However, neural avalanches have been found to occur in a wide the notion of a critical state corresponds to a point in range of neural tissues and species [4]. From a theoreti- the control parameter space of a system. Assuming here cal point of view, it has been suggested [4, 5] that such the cholinergic coupling as a control parameter, the above a scale-free organisation could foster information storage narrative is incompatible with the framework of critical- and transfer, improvement of the computational capabil- ity. In contrast, Zheng et al observations would suggest ities [6], information transmission [3, 6, 7], sensitivity to that criticality, if any, (a) either lasts for a short mo- stimuli and enlargement of dynamic range [8{11]. ment during development where the coupling parameter In this spirit it has been proposed by M. Hennig et al, is right at a critical value, ruining any hope of robustness in a paper combining experiments and modelling, that of the phenomenon, or (b) occurs within an interval of such a critical regime could also take place in the vi- this coupling parameter. sual system, at the early stages of its development [12]. Two theoretical arguments could solve this apparent Investigating the dynamics of stage II retinal waves, a incompatibility: (i) There is a hidden mechanism adapt- mechanism participating in the development of the vi- ing to the coupling parameter variations so as to maintain sual system of mammals [13{16] (see section II A for a the retina in a critical state, in a mechanism reminiscent detailed description), they show that the network of neu- to self-organized criticality (SOC) e.g. the scenario called rons (here Starburst Amacrine Cells - SACs) is capable "Mapping Self-Organized Criticality onto Criticality" in- of operating at a transition point between purely local troduced by D. Sornette and co-workers in [18]. In the arXiv:1912.03934v1 [q-bio.NC] 9 Dec 2019 and global functional connectedness, corresponding to a past, this line of thought has collected a broad consen- percolation phase transition, where waves of activity - of- sus and resulted in numerous interesting results [18{23]. ten referred to as "avalanches"- are distributed according For example, in [19] and more recently in [21], dynamical to power laws (see Fig. 4 of [12]). They interpret this synapses are shown to be responsible for self-organized regime as an indication that early spontaneous activity in criticality. In [24], the feedback of the control param- the developing retina is regulated according to the follow- eter to the order parameter is carried out through the ing principle; maximize randomness and variability in the dynamics of the synaptic tree radius. (ii) Retinal waves resulting activity patterns. This remark is in complete dynamics indeed displays power law distributions on an agreement with the idea of dynamic range maximization interval of the coupling parameter, without the need of [17] and could be of central importance for our under- adding an extra mechanism. standing of the visual system. In fact, it suggests that, In this paper we report on the action of noise fluctu- during its formation, the visual system could be driven by ations onto the retinal waves dynamics and give strong spontaneous events, namely the retinal waves, exhibiting arguments in favor of hypothesis (ii). In the past, noise the characteristics of a second order phase transition. has already be shown to lead to unexpected behavior in 2 excitable systems. In 1997, studying the dynamics of a the geometry of waves dynamics and their potential links single excitable FitzHugh-Nagumo neuron under exter- to visual system development. nal noise driving, Pikovsky and Kurths [25] reported the existence of a coherence resonance mechanism, charac- terised by the existence of a noise amplitude for which the II. STAGE II RETINAL WAVES MODEL self-correlation characteristic time displays a maximum. Moreover, for this noise amplitude, the signal to noise A. The biophysics of retinal waves ratio displays a minimum. Later, on neural networks cell cultures and in computer simulations, the existence of an Retinal waves are bursts of activity occurring spon- optimal level of noise for which the regularity of the syn- taneously in the developing retina of vertebrate species, chronized neural interburst interval becomes maximal, contributing to the shaping of the visual system organiza- has been reported [26]. tion [13{16]. They are characterized by localized groups Here, we argue that noise is responsible for the broad- of neurons becoming simultaneously active, initiated at ening of the region of cholinergic coupling where stage II random points and propagating at speeds ranging from retinal waves exhibit power laws. Starting from a model 100µm/s (mouse, [31], [32]) up to 400µm/s (chick, [33]), we developed in [1] describing the individual bursting be- with changing boundaries, dependent on local refractori- havior of Starburst Amacrine Cells (SACs) and reproduc- ness [16, 34]. This activity, slowly spreading across the ing numerous experimental observations, we study the retina, is an inherent property of the retinal network [35]. properties of stage II retinal waves and their dependence More precisely, the generation of waves requires three upon the cholinergic coupling. The existence of a phase conditions [16, 36]: transition between asynchronized and synchronized pulse coupled excitable oscillators has already been well estab- (C1) A source of depolarization for wave initiation ("How lished in the literature [27{29]. We do observe it as well as do waves start?"). Given that there is no exter- the usual power law behaviours are commonly observed nal input (e.g. from visual stimulation in the early at the threshold of phase transition. But here, because of retina), there must be some intrinsic mechanism by the system's closeness to a saddle node bifurcation point, which neurons become active; thoroughly analysed in [1], retinal waves can be initiated by noise by a mechanism detailed below. This leads to a (C2) A network of excitatory interactions for propaga- surprising consequence: in the absence of any SOC-like tion ("How do waves propagate?"). Once some neu- mechanism we numerically show the existence of a whole rons become spontaneously active, how do they ex- interval of cholinergic coupling where retinal waves ex- cite neighboring neurons ? hibit power laws. The width of this interval depends on (C3) A source of inhibition that limits the spatial ex- noise and there is an optimal noise level where this inter- tent of waves and dictates the minimum interval val has a maximal width. We call this effect noise driven between them (How do waves stop?). broadening of the neural synchronisation transition. The paper is organised as follows. In section II we Wave activity begins in the early development, long be- briefly outline the conductance-based dynamical model fore the retina is responsive to light. It emerges due previously used in [1] to describe the individual SAC dy- to several biophysical mechanisms which change during namics and quickly remind the main associated results. development, dividing retinal waves maturation into 3 We then introduce the synaptic cholinergic coupling, dis- stages (I, II, III) [15]. Each stage, mostly studied in cuss the complexity of the ensuing model and the vari- mammals, is characterized by a certain type of network ous difficulties associated with its numerical simulation interaction (condition C2): gap junctions for stage I; and conclude with the validity limits of our numerical cholinergic transmission for stage II; and glutamatergic investigation. Section III deals with the neural synchro- transmission for stage III. In this work, we focus on stage nisation phase transition. Numerical evidence of such a II. phase transition is given from the point of view of global During this period, the principal mechanism for trans- firing rate (i.e. the total number of spiking SACs at a mission is due to the neurotransmitter acetylcholine.