• Bernoulli
  • Volume 25, Number 1 (2019), 771-792.

Estimating the interaction graph of stochastic neural dynamics

Aline Duarte, Antonio Galves, Eva Löcherbach, and Guilherme Ost

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In this paper, we address the question of statistical model selection for a class of stochastic models of biological neural nets. Models in this class are systems of interacting chains with memory of variable length. Each chain describes the activity of a single neuron, indicating whether it spikes or not at a given time. The spiking probability of a given neuron depends on the time evolution of its presynaptic neurons since its last spike time. When a neuron spikes, its potential is reset to a resting level and postsynaptic current pulses are generated, modifying the membrane potential of all its postsynaptic neurons. The relationship between a neuron and its pre- and postsynaptic neurons defines an oriented graph, the interaction graph of the model. The goal of this paper is to estimate this graph based on the observation of the spike activity of a finite set of neurons over a finite time. We provide explicit exponential upper bounds for the probabilities of under- and overestimating the interaction graph restricted to the observed set and obtain the strong consistency of the estimator. Our result does not require stationarity nor uniqueness of the invariant measure of the process.

Article information

Bernoulli, Volume 25, Number 1 (2019), 771-792.

Received: April 2016
Revised: June 2017
First available in Project Euclid: 12 December 2018

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Zentralblatt MATH identifier

biological neural nets graph of interactions interacting chains of variable memory length statistical model selection


Duarte, Aline; Galves, Antonio; Löcherbach, Eva; Ost, Guilherme. Estimating the interaction graph of stochastic neural dynamics. Bernoulli 25 (2019), no. 1, 771--792. doi:10.3150/17-BEJ1006.

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  • [1] Adrian, E. and Bronk, D. (1929). The discharge of impulses in motor nerve fibres. Part II. The frequency of discharge in reflex and voluntary contractions. J. Physiol. 67 119–151.
  • [2] Adrian, E.D.A. (1928). The Basis of Sensation: The Action of the Sense Organs. London: Christophers.
  • [3] Bresler, G. (2015). Efficiently learning Ising models on arbitrary graphs. In Proceedings of the Forty-Seventh Annual ACM on Symposium on Theory of Computing 771–782. New York, NY, USA: ACM.
  • [4] Bresler, G., Mossel, E. and Sly, A. (2008). Reconstruction of Markov random fields from samples: Some observations and algorithms. In Proceedings of the 11th International Workshop, APPROX 2008, and 12th International Workshop, RANDOM 2008 on Approximation, Randomization and Combinatorial Optimization: Algorithms and Techniques 343–356. Berlin, Heidelberg: Springer.
  • [5] Brillinger, D.R. (1988). Maximum likelihood analysis of spike trains of interacting nerve cells. Biol. Cybernet. 59 189–200.
  • [6] Brillinger, D.R. and Segundo, J.P. (1979). Empirical examination of the threshold model of neuron firing. Biol. Cybernet. 35 213–220. DOI:10.1007/BF00344204.
  • [7] Brochini, L., Costa, A.A., Abadi, M., Roque, A.C., Stolfi, J. and Kinouchi, O. (2016). Phase transitions and self-organized criticality in networks of stochastic spiking neurons. Sci. Rep. 6 35831. DOI:10.1038/srep35831.
  • [8] Brochini, L., Hodara, P., Pouzat, C. and Galves, A. (2017). Interaction graph estimation for the first olfactory relay of an insect. Available at arXiv:1612.05226.
  • [9] Csiszár, I. and Talata, Z. (2006). Consistent estimation of the basic neighborhood of Markov random fields. Ann. Statist. 34 123–145. DOI:10.1214/009053605000000912.
  • [10] De Masi, A., Galves, A., Löcherbach, E. and Presutti, E. (2015). Hydrodynamic limit for interacting neurons. J. Stat. Phys. 158 866–902. DOI:10.1007/s10955-014-1145-1.
  • [11] Duarte, A. and Ost, G. (2016). A model for neural activity in the absence of external stimulus. Markov Process. Related Fields 22 37–52.
  • [12] Duarte, A., Ost, G. and Rodríguez, A.A. (2015). Hydrodynamic limit for spatially structured interacting neurons. J. Stat. Phys. 161 1163–1202. DOI:10.1007/s10955-015-1366-y.
  • [13] Dyan, P. and Abbott, L.F. (2001). Theoretical Neuroscience. Computational and Mathematical Modeling of Neural Systems. MIT Press.
  • [14] Fournier, N. and Löcherbach, E. (2016). On a toy model of interacting neurons. Ann. Inst. Henri Poincaré Probab. Stat. 52 1844–1876.
  • [15] Galves, A. and Leonardi, F. (2008). Exponential inequalities for empirical unbounded context trees. In In and Out of Equilibrium. 2. Progress in Probability 60 257–269. Birkhäuser, Basel.
  • [16] Galves, A. and Löcherbach, E. (2013). Infinite systems of interacting chains with memory of variable length – A stochastic model for biological neural nets. J. Stat. Phys. 151 896–921.
  • [17] Galves, A., Orlandi, E. and Takahashi, D.Y. (2015). Identifying interacting pairs of sites in Ising models on a countable set. Braz. J. Probab. Stat. 29 443–459.
  • [18] Gerstner, W. (1995). Time structure of the activity in neural network models. Phys. Rev. E 51 738–758. DOI:10.1103/PhysRevE.51.738.
  • [19] Gerstner, W. and Kistler, W.M. (2002). Spiking Neuron Models: Single Neurons, Populations, Plasticity. Cambridge: Cambridge Univ. Press.
  • [20] Gerstner, W. and van Hemmen, J.L. (1992). Associative memory in a network of spiking neurons. Network 3 139–164.
  • [21] Hodara, P. and Löcherbach, E. (2017). Hawkes processes with variable length memory and an infinite number of components. Adv. in Appl. Probab. 49 84–107.
  • [22] Hodgkin, A.L. and Huxley, A.F. (1952). A quantitative description of membrane current and its application to conduction and excitation in nerve. J. Physiol. 117 500–544.
  • [23] Lerasle, M. and Takahashi, D.Y. (2011). An oracle approach for interaction neighborhood estimation in random fields. Electron. J. Stat. 5 534–571.
  • [24] Lerasle, M. and Takahashi, D.Y. (2016). Sharp oracle inequalities and slope heuristic for specification probabilities estimation in discrete random fields. Bernoulli 22 325–344.
  • [25] Löcherbach, E. and Orlandi, E. (2011). Neighborhood radius estimation for variable-neighborhood random fields. Stochastic Process. Appl. 121 2151–2185.
  • [26] Montanari, A. and Pereira, J.A. (2009). Which graphical models are difficult to learn? In Advances in Neural Information Processing Systems 22 1303–1311. Curran Associates, Inc.
  • [27] Ravikumar, P., Wainwright, M.J. and Lafferty, J.D. (2010). High-dimensional Ising model selection using $\ell_{1}$-regularized logistic regression. Ann. Statist. 38 1287–1319.
  • [28] Reynaud-Bouret, P., Rivoirard, V. and Tuleau-Malot, C. (2013). Inference of functional connectivity in neurosciences via Hawkes processes. In 1st IEEE Global Conference on Signal and Information Processing.
  • [29] Robert, P. and Touboul, J. (2016). On the dynamics of random neuronal networks. J. Stat. Phys. 165 545–584.
  • [30] Soudry, D., Keshri, S., Stinson, P., Oh, M.-H., Iyengar, G. and Paninski, L. (2015). Efficient “shotgun” inference of neural connectivity from highly sub-sampled activity data. PLoS Comput. Biol. 11 e1004464.
  • [31] Talata, Z. (2014). Markov neighborhood estimation with linear complexity for random fields. In Information Theory (ISIT), 2014 IEEE International Symposium on 3042–3046.
  • [32] Yaginuma, K. (2016). A stochastic system with infinite interacting components to model the time evolution of the membrane potentials of a population of neurons. J. Stat. Phys. 163 642–658.