Simulation Based Network Structure Inference Constrained by Observed Spike Trains: SIMBADNESTICOST ANR-22-CE45-0027

Neurophysiologists are nowadays able to record from a large number of extracellular electrodes and to extract, from the raw data, the sequences of action potentials or spikes generated by many neurons. Unfortunately these "many neurons" still represent only a tiny fraction of the neuronal population which constitutes the network. Using association statistics such as the estimation of the cross-correlation functions, they try and infer the structure of the network formed by the recorded neurons. But this inference is compromised by the tremendous under-sampling of the neuronal population and by the errors made during the sequences reconstruction. This yields a "network picture" usually called a "functional network" whose features depend strongly on the recording conditions (such as the presence/absence of a stimulation). We consider that reconstructing the network formed by the recorded neurons is an ill-posed problem. We propose to focus instead on the "generative probability distribution" of the network: what is the probability to have a connection from a type A neuron to a type B neuron? Is the probability to have a connection from neuron Y of type B to neuron X of type A dependent on the presence of a connection from X to Y? We propose to simulate first the whole network using a simplified neuronal dynamics and different (parametrized) generative probability distributions. We will then compare the association statistics between the simulated and the experimentally observed cases. This type of approach is now commonly used in several fields under different names like "Approximate Bayesian Computation" or "Simulation based Inference". We will then be able to asses if there is an "over representation of reciprocal connections" using data from the first olfactory relay of an insect.

This Project runs from January 2023 to December 2026 and is funded by the ANR (ANR-22-CE45-0027) in the program AAPG2022.

A team from the Institut de Recherche Mathématique Avancée (IRMA) in the University of Strasbourg:

• Christophe Pouzat
• Ségolen Geffray
• Étienne Birmelé
• Samuel Maistre
• Emmanuel Franck
• Vincent Vigon
• Louise Martineau

We are going to give full details on how we implement the project as publications proceed, but the key ideas were already plainly stated in:

• Diggle and Gratton (1984) Monte Carlo Methods of Inference for Implicit Statistical Models, Journal of the Royal Statistical Society: Series B (Methodological), Volume 46, Issue 2, Pages 193–212.
• Wood (2010) Statistical inference for noisy nonlinear ecological dynamic systems, Nature, 466, Pages 1102–1104.

For the impatient, what we do now is very close to what is done in the last example of Diggle and Gratton (1984). We use the antennal lobe (first olfactory relay) of the locust, Schistocerca americana as a "test" system. A lot of data are publicly available for this system on zenodo. This network is made of 810 Projection neurons that spike and about 300 interneurones that do not spike. We choose a probability distribution constrained by known anatomical data for the graph (this can be seen as a rule that allows us to generate networks by drawing connections with probabilities that are specific to the types of neurons connected; these probabilities are in fact the model parameters we want to estimate), we choose a dynamics for the neurons (this dynamics also depends on some parameters that are drawn from a prior distribution), we simulate the whole network, we compute statistics (inter-spike interval distributions, cross-correlograms, etc.) from the simulated spike trains and we estimate the distribution of these statistics, before comparing these distributions with the (same) statistics computed from the real data.

1. Antonio Galves, Eva Löcherbach and Christophe Pouzat (2024) Probabilistic Spiking Neuronal Nets. Neuromathematics for the Computer Era, Springer, book series: Lecture Notes on Mathematical Modelling in the Life Sciences. The companion website describes and documents the codes simulating the models introduced in the book.
2. Morgan André, Christophe Pouzat (2025) A Quasi-Stationary Approach to Metastability in a System of Spiking Neurons with Synaptic Plasticity, Mathematical Neuroscience and Applications 4, DOI: 10.46298/mna.7668; the associated simulation and analysis codes are available from the following GitLab repository: Metastability in a System of Spiking Neurons with Synaptic Plasticity.

• Loi quasi-stationnaire et méta-stabilité pour un modèle neuronal stochastique de la mémoire de travail, présentation donnée au séminaire Math-Bio-Santé à Toulouse le 13 septembre 2024. Tous les détails et toutes les références se trouvent dans le manuscrit déposé sur HAL.
• A Quasi-Stationary Approach to Metastability in a System of Spiking Neurons with Synaptic Plasticity, joint work with Morgan André: an Invited Lecture given at LASCON IX, on January 22 2024 in the Special Session in Honor of Antonio Galves. The talk is available on YouTube.
• What to do with extracellular recordings? A proposal. A talk given at the Eurandom's Stochastic Models in Life Science workshop on September 6 2023.
• Neuronal network structure inference by simulation, MoMA seminar, Dipartimento di Matematica, Sapienza, Università di Roma, May 19 2023.

This is work in progress!

• Probabilistic Spiking Neuronal Nets: Companion contains mostly `Python` codes illustrating how to simulated the models of the book with the same title and how to extend these models.
• network-codes-in-fortran a Fortran version of the codes for more "serious" applications.