Noisy spiking neurons and networks: useful approximations for firing probabilities and global behavior.

Electrophysiological properties of spiking neurons receiving complex stimuli perturbed by noise are investigated. A semi-analytical estimate of firing probabilities and subthreshold behavior of the stochastic system can be made in terms of the solution of a purely deterministic system. The method comes from an approximation for the distribution function and moments of the underlying non linear multidimensional diffusion process. This so called moment method works for general conductance-based systems and an application is presented for the Hodgkin-Huxley neuronal model. Statistical properties obtained from the moment method are compared with direct numerical integration of the stochastic system. The firing probability due to external noise is derived as a closed formula. Results are given for different forms of the deterministic component of the stimulus. A generalization to neural networks of conductance-based systems with internal currents perturbed by noise can be obtained using the same approach. In the case of fully connected networks, a mean field population equation is derived which may be compared to Kuramoto's master equation for weakly coupled neural oscillators.

[1]  P Blanchard,et al.  Stochastic dynamical aspects of neuronal activity , 1993, Journal of mathematical biology.

[2]  H Lecar,et al.  Theory of threshold fluctuations in nerves. I. Relationships between electrical noise and fluctuations in axon firing. , 1971, Biophysical journal.

[3]  Klaus Schulten,et al.  Effect of noise and perturbations on limit cycle systems , 1991 .

[4]  H. Tuckwell,et al.  Statistical properties of stochastic nonlinear dynamical models of single spiking neurons and neural networks. , 1996, Physical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topics.

[5]  O. Prospero-Garcia,et al.  Reliability of Spike Timing in Neocortical Neurons , 1995 .

[6]  Henry C. Tuckwell.  Random Perturbations of the Reduced Fitzhugh-Nagumo Equation , 1992 .

[7]  J. M. Ritchie,et al.  A quantitative description of membrane currents in rabbit myelinated nerve. , 1979, The Journal of physiology.

[8]  Henry C. Tuckwell,et al.  Stochastic processes in the neurosciences , 1989 .

[9]  Roger Rodriguez.  Coupled Hodgkin Huxley Neurons with Stochastic Synaptic Inputs , 1995 .

[10]  J J Hopfield,et al.  Rapid local synchronization of action potentials: toward computation with coupled integrate-and-fire neurons. , 1995, Proceedings of the National Academy of Sciences of the United States of America.

[11]  G. Ermentrout,et al.  Oscillator death in systems of coupled neural oscillators , 1990 .

[12]  Y. Kuramoto.  Collective synchronization of pulse-coupled oscillators and excitable units , 1991 .

[13]  Germán Mato,et al.  Synchrony in Excitatory Neural Networks , 1995, Neural Computation.

[14]  A. Hodgkin,et al.  A quantitative description of membrane current and its application to conduction and excitation in nerve , 1952, The Journal of physiology.

[15]  Marius Usher,et al.  Dynamics of Populations of Integrate-and-Fire Neurons, Partial Synchronization and Memory , 1993, Neural Computation.

[16]  J. Gazeau,et al.  Modern Group Theoretical Methods in Physics , 1995 .

[17]  William R. Softky,et al.  The highly irregular firing of cortical cells is inconsistent with temporal integration of random EPSPs , 1993, The Journal of neuroscience : the official journal of the Society for Neuroscience.

[18]  H. L. Bryant,et al.  Spike initiation by transmembrane current: a white‐noise analysis. , 1976, The Journal of physiology.

[19]  J. Rinzel,et al.  Clustering in globally coupled inhibitory neurons , 1994 .

[20]  G L Gerstein,et al.  Detecting spatiotemporal firing patterns among simultaneously recorded single neurons. , 1988, Journal of neurophysiology.

[21]  A. Jazwinski.  Stochastic Processes and Filtering Theory , 1970 .

[22]  H Lecar,et al.  Squid axon membrane response to white noise stimulation. , 1974, Biophysical journal.