CNS*2025 Florence

P302 Is the Cortical dynamics ergodic?

Ferdinand Tixidre*1, , Gianluigi Mongillo2,3, Alessandro Torcini1
● Laboratoire de Physique Théorique et Modélisation, CY Cergy Paris Université, Cergy-Pontoise, France
● School of Natural Sciences, Institute for Advanced Study, Princeton, NJ, USA.
● Institut de la Vision, Sorbonne Université, Paris, France.


*Email: ferdinand.tixidre@cyu.fr

Introduction

Cortical neuronsin vivoshow significant temporalvariability in their spike trains even in virtually-identicalexperimental conditions. This variability is partly due to the intrinsic stochasticity of the spike-generation. To accountforthe levels of variabilityobserved, one needs to assume additional fluctuations in the activity over longer timescales [1, 2]. But what is their origin? One theory suggest they result from non-ergodic network dynamics [3] arising from partially-symmetric synaptic connectivity, consistently with anatomical observations [4]. However it is unclear, if such ergodicity breakingoccurs in networks of spiking neurons, due to fast temporal fluctuations in the synaptic inputs[5].


Methods
To address these questions, we study sparsely-connected networks of inhibitory leaky-integrate-and-fire neurons with arbitrary levels of symmetry, q, in the synaptic connectivity. The connectivity matrix ranges from random (q=0) to fully symmetric (q=1). Neurons also receive a constant excitatory drive, balanced by recurrent synaptic inputs. To assess ergodicity, we estimate single-neuron firing rates over increasing time intervals, T, starting from different initial membrane voltage distributions (for the same network). If the dynamics is ergodic, the difference, D, between estimates from different initial conditions should approach zero as 1/T for large T.

Results
This is, in fact, what happens in random networks (i.e., q = 0; Fig. 1(a)). In partially-symmetric networks (q >0), the onset of the "ergodic" regime occurs at longer and longer times. The situation becomes dramatic for the fully symmetric network (q= 1), where D does not decay even for time windows that are 5 order of magnitudes longer than the membrane time constant as shown in Fig 1(a); the network dynamics is non-ergodic, at least in a weak sense. In this regime, the network activity is sparse, with a large fraction of almost-silent neurons, and the auto-covariance function of the spike trains exhibits long time scales (Fig. 1 (b)). Both these features are also routinely observed in experimental recordings [6,7]

Discussion
Taken together, our results provide support to the idea that many features of cortical activity can be parsimoniously explained by the non-ergodicity of the network dynamics. In particular, in this regime, the activity level of the single neurons can significantly change depending on the “microscopic" initial conditions (which are beyond experimental control) (Fig. 1 (c-d)), providing a simple explanation for the large trial-to-trial fluctuations observed in the experiments.





Figure 1. (a) D as a function of time for different values of q: q=0 (blue); q=0.5 (green); q=0.8 (orange); q=0.9 (red); q=0.95 (brown); q=1.0 (black). (b) Auto-correlation of synaptic currents for different q. (c-d) Cumulative firing rate of a single neuron for q=0.8 (c) and q=0.9 (d). Shades of the main color represent different replicas. The insets show the instantaneous firing rate of the same neuron.
Acknowledgements
F.T. and A.T. received financial support by the Labex MME-DII (Grant No. ANR-11-LBX-0023-01) and by CY Generations
(Grant No ANR-21-EXES-0008). G.M. work is supported by grants ANR-19-CE16-0024-01 and ANR-20-CE16-0011-02 from the French National Research Agency and by a grant from the Simons Foundation (891851, G.M.).


References
● https://doi.org/10.1016/j.neuron.2010.12.037
● https://doi.org/10.1167/18.8.8
● https://www.biorxiv.org/cgi/content/short/2022.03.14.484348
● https://doi.org/10.1126/science.abj5861
● https://doi.org/10.1038/s41598-019-40183-8
● https://doi.org/10.1007/s00359-006-0117-6
● https://doi.org/10.1038/nn.3862