Skip to main content

On Stationary Distributions of Stochastic Neural Networks

  • K. Borovkov (a1), G. Decrouez (a1) and M. Gilson (a2)

The paper deals with nonlinear Poisson neuron network models with bounded memory dynamics, which can include both Hebbian learning mechanisms and refractory periods. The state of the network is described by the times elapsed since its neurons fired within the post-synaptic transfer kernel memory span, and the current strengths of synaptic connections, the state spaces of our models being hierarchies of finite-dimensional components. We prove the ergodicity of the stochastic processes describing the behaviour of the networks, establish the existence of continuously differentiable stationary distribution densities (with respect to the Lebesgue measures of corresponding dimensionality) on the components of the state space, and find upper bounds for them. For the density components, we derive a system of differential equations that can be solved in a few simplest cases only. Approaches to approximate computation of the stationary density are discussed. One approach is to reduce the dimensionality of the problem by modifying the network so that each neuron cannot fire if the number of spikes it emitted within the post-synaptic transfer kernel memory span reaches a given threshold. We show that the stationary distribution of this ‘truncated’ network converges to that of the unrestricted network as the threshold increases, and that the convergence is at a superexponential rate. A complementary approach uses discrete Markov chain approximations to the network process.

Corresponding author
Postal address: Department of Mathematics and Statistics, The University of Melbourne, Parkville, VIC 3010, Australia.
∗∗ Email address:
∗∗∗ Current address: Departament de Tecnologies de la Informació i les Comunicacions, Universitat Pompeu Fabra, Barcelona 08018, Spain.
Hide All
[1] Bear, M. F., Connors, B. W. and Paradiso, M. A. (2007). Neuroscience: Exploring the Brain, 3rd edn. Lippincott Williams & Wilkins, Philadelphia, PA.
[2] Borovkov, A. A. (1998). Ergodicity and Stability of Stochastic Processes. John Wiley, Chichester.
[3] Borovkov, A. A. (2013). Probability Theory. Springer, London.
[4] Borovkov, K. and Last, G. (2008). On level crossings for a general class of piecewise-deterministic Markov processes. Adv. Appl. Prob. 40, 815834.
[5] Borovkov, K., Decrouez, G. and Gilson, M. (2012). On stationary distributions of stochastic neural networks. Preprint. Available at
[6] Brémaud, P. (1981). Point Processes and Queues. Springer, New York.
[7] Brémaud, P. and Massoulié, L. (1996). Stability of nonlinear Hawkes processes. Ann. Prob. 24, 15631588.
[8] Brémaud, P. and Massoulié, L. (2001). Hawkes branching point processes without ancestors. J. Appl. Prob. 38, 122135.
[9] Brémaud, P. and Massoulié, L. (2002). Power spectra of general shot noises and Hawkes point processes with a random excitation. Adv. Appl. Prob. 34, 205222,
[10] Brillinger, D. R. (1975). The identification of point process systems. Ann. Prob. 3, 909929.
[11] Brillinger, D. R. (1988). Maximum likelihood analysis of spike trains of interacting nerve cells. Biol. Cybernetics 59, 189200.
[12] Burkitt, A. N., Gilson, M. and van Hemmen, J. L. (2007). Spike-timing-dependent plasticity for neurons with recurrent connections. Biol. Cybernetics 96, 533546.
[13] Chornoboy, E. S., Schramm, L. P. and Karr, A. F. (1988). Maximum likelihood identification of neural point process systems. Biol. Cybernetics 59, 265275.
[14] Dayan, P. and Abbott, L. F. (2001). Theoretical Neuroscience. Computational and Mathematical Modeling of Neural Systems. MIT Press, Cambridge, MA.
[15] Gerstner, W. and Kistler, W. M. (2002). Spiking Neuron Models. Single Neurons, Populations, Plasticity. Cambridge University Press.
[16] Gilson, M. (2009). Biological learning mechanisms in spiking neuronal networks. , The University of Melbourne. Available at
[17] Gilson, M. et al. (2009). Emergence of network structure due to spike-timing-dependent plasticity in recurrent neuronal networks III: Partially connected neurons driven by spontaneous activity. Biol. Cybernetics 101, 411426.
[18] Gilson, M. et al. (2009). Emergence of network structure due to spike-timing-dependent plasticity in recurrent neuronal networks IV. Biol. Cybernetics 101, 427444.
[19] Gilson, M., Burkitt, A. and van Hemmen, J. L. (2010). STDP in recurrent neuronal networks. Front. Comput. Neurosci. 4, 23.
[20] Harrison, J. M. and Resnick, S. I. (1976). The stationary distribution and first exit probabilities of a storage process with general release rule. Math. Operat. Res. 1, 347358.
[21] Hawkes, A. G. (1971). Point spectra of some mutually exciting point processes. J. R. Statist. Soc. B 33, 438443.
[22] Izhikevich, E. M. (2007). Dynamical Systems in Neuroscience: The Geometry of Excitability and Bursting. MIT Press, Cambridge, MA.
[23] Jacobsen, M. (2006). Point Process Theory and Applications: Marked Point and Piecewise Deterministic Processes. Birkhäuser, Boston, MA.
[24] Kempter, R., Gerstner, W. and van Hemmen, J. L. (1999). Hebbian learning and spiking neurons. Phys. Rev. E 59, 44984514.
[25] Koch, C. and Idan, S. (eds) (1998). Methods in Neuronal Modeling: From Ions to Networks, 2nd edn. MIT Press, Cambridge, MA.
[26] Massoulié, L. (1998). Stability results for a general class of interacting point processes dynamics, and applications. Stoch. Process. Appl. 75, 130.
[27] Nicholls, J. G. et al. (2012). From Neuron to Brain, 5th edn. Sinauer Associates, Sunderland, MA.
[28] Paninski, L. (2004). Maximum likelihood estimation of cascade point-process neural encoding models. Network: Comput. Neural Systems 15, 243262.
[29] Pillow, J. W., Ahmadian, Y. and Paninski, L. (2011). Model-based decoding, information estimation, and change-point detection techniques for multineuron spike trains. Neural Comput. 23, 145.
[30] Pillow, J. W. et al. (2008). Spatio-temporal correlations and visual signalling in a complete neuronal population. Nature 454, 995999.
[31] Shepherd, G. M. (ed.) (2004). The Synaptic Organization of the Brain, 5th edn. Oxford University Press.
[32] Shepherd, G. M. and Grillner, S. (eds) (2010). Handbook of Brain Microcircuits. Oxford University Press.
[33] Sporns, O. (2011). Networks of the Brain. MIT Press, Cambridge, MA.
[34] Stevenson, I. H. et al. (2009). Bayesian inference of functional connectivity and network structure from spikes. IEEE Trans. Neural Systems Rehabil. Eng. 17, 203213.
[35] Truccolo, W. et al. (2005). A point process framework for relating neural spiking activity to spiking history, neural ensemble, and extrinsic covariate effects. J. Neurophysiol. 93, 10741089.
Recommend this journal

Email your librarian or administrator to recommend adding this journal to your organisation's collection.

Journal of Applied Probability
  • ISSN: 0021-9002
  • EISSN: 1475-6072
  • URL: /core/journals/journal-of-applied-probability
Please enter your name
Please enter a valid email address
Who would you like to send this to? *


MSC classification


Full text views

Total number of HTML views: 0
Total number of PDF views: 0 *
Loading metrics...

Abstract views

Total abstract views: 0 *
Loading metrics...

* Views captured on Cambridge Core between <date>. This data will be updated every 24 hours.

Usage data cannot currently be displayed