P082 Temporal Dynamics of Inter-Spike Intervals in Neural Populations
Luca Falorsi*1,2, Gianni v. Vinci2, Maurizio Mattia2
1PhD program in Mathematics, Sapienza Univ. of Rome, Piazzale Aldo Moro 5, Rome, Italy
2Natl. Center for Radiation Protection and Computational Physics, Istituto Superiore di Sanità, Viale Regina Elena 299, Rome, Italy
*Email: luca.falorsi@gmail.com
Introduction
The study of inter-spike interval (ISI) distributions in neuronal populations plays a crucial role in linking theoretical models with experimental data [1, 2]. As an experimentally accessible measure, ISI distributions provide critical insights into how neurons code and process information [3–5]. However, characterizing these distributions in populations of spiking neurons far from equilibrium remains an open issue. In this work, we develop a population density framework [6–8] to study the joint dynamics of the time from the last spike (τ) and the membrane potential (v) in homogeneous networks of integrate-and-fire neurons.
Methods
We model the network dynamics using a population density approach, where a joint probability distribution describes the fraction of neurons with membrane potential (v) and elapsed time (τ) since their last spike. This distribution evolves according to a two-dimensional Fokker-Planck partial differential equation (PDE), allowing us to systematically analyze how single-neuron ISI distributions change over time, including nonstationary conditions driven by external inputs or network interactions. To further characterize ISI statistics, we derive a hierarchy of one-dimensional PDEs describing the evolution of ISI moments and analytically study first-order perturbations from the stationary state, providing first-order corrections to renewal theory.
Results
As a first step, we analytically solve the relaxation dynamics towards the steady state for an uncoupled population of neurons, obtaining an explicit expression for the time-dependent ISI. We then show, through numerical simulations, that the introduced equation correctly captures the time evolution of the ISI distribution, even when the population significantly deviates from its stationary state, such as in the presence of limit cycles or time-varying external stimuli (Fig. 1). Additionally, by self-consistently incorporating the sampled empirical firing rate, the resulting stochastic Fokker-Planck equation describes finite-size fluctuations. Spiking network simulations show an excellent agreement with the numerical integration of the PDE.
Discussion
We connect our novel population density approach to the Spike Response Model (SRM) [10], demonstrating that marginalizing over v recovers the Refractory Density Method (RDM) [11]. However, the marginal equation remains unclosed, and both SRM and RDM rely on a quasi-renewal approximation based on the stationary ISI distribution.
In conclusion, we developed an analytic framework to characterize ISI distributions in nonstationary regimes. Our approach, validated through simulations, bridges theoretical models with experimental observations. Furthermore, this work paves the way for analytically studying synaptic plasticity mechanisms that depend on the timing of the last spike, such as spike-timing-dependent plasticity.
Figure 1. ISI dynamics in an excitatory limit cycle (same parameters as [9] ). Comparing Spiking Neural Network simulations (SNN) with Fokker-Planck equation (FP) and its stochastic version (SFP). Time is measured in units of the membrane time constant τ_m=20ms. A Phase-dependent ISI distribution. B Trajectory of the firing rate and the first moment of the ISI. C Time averaged ISI distribution.
Acknowledgements
LF aknowledges support by ICSC – Centro Nazionale di Ricerca in High Performance Computing, Big Data and Quantum Computing and Sapienza University of Rome (AR12419078A2D6F9).
MM and GV acknowledge support from the Italian National Recovery and Resilience Plan (PNRR), M4C2, funded by the European Union–NextGenerationEU (Project IR0000011, CUP B51E22000150006, “EBRAINS-Italy”)
References
1.https://doi.org/10.1016/s0006-3495(64)86768-0
2.https://doi.org/10.2307/3214232
3.https://doi.org/10.1523/JNEUROSCI.13-01-00334.1993
4.https://doi.org/10.1103/PhysRevLett.67.656
5.https://doi.org/10.1523/JNEUROSCI.18-10-03870.1998
6.https://doi.org/10.1162/089976699300016179
7.https://doi.org/10.1162/089976600300015673
8.https://doi.org/10.1103/PhysRevE.66.051917|
9.https://doi.org/10.1103/PhysRevLett.130.097402
10.https://doi.org/10.1103/PhysRevE.51.738
11.https://doi.org/10.1016/j.conb.2019.08.003
