# The maximum likelihood approach to the identification of neuronal firing systems

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## Abstract

The concern of this work is the identification of the (nonlinear) system of a neuron firing under the influence of a continuous input in one case, and firing under the influence of two other neurons in a second case. In the first case, suppose that the data consist of sample values X where where γ with X

_{t}, Y_{t}, t=0, ±1, ±2, ... with Y_{t}=1 if the neuron fires in the time interval t to t+1 and Y_{t}=0 otherwise, and with X_{t}denoting the (sampled) noise value at time t. Suppose that H_{t}denotes the history of the process to time t. Then, in this case the model fit has the form$$Prob\{ Y_t = 1|H_t \} = \Phi (U_t - \theta )$$

$$U_t = \sum\limits_{u = 0}^{\gamma _t - 1} {a_u } X_{t - u} + \sum\limits_{u = 0}^{\gamma _t - 1} { \sum\limits_{\nu = 0}^{\gamma _t - 1} {b_{u,\nu } } X_{t - u} X_{t - \nu } } $$

_{ t }denotes the time elapsed since the neuron last fired and ϕ denotes the normal cumulative. This model corresponds to quadratic summation of the stimulus followed by a random threshold device. In the second case, a network of three neurons is studied and it is supposed that$$U_t = \sum\limits_{u = 0}^{\gamma _t - 1} {a_u } X_{t - u} + \sum\limits_{u = 0}^{\gamma _t - 1} {b_u } X_{t - u} $$

_{t}and Z_{t}zero-one series corresponding to the firing times of the two other neurons. The models are fit by the method of maximum likelihood to*Aplysia californica*data collected in the laboratory of Professor J.P. Segundo. The paper also contains some general comments of the advantages of the maximum likelihood method for the identification of nonlinear systems.## Keywords

Causal connections Maximum likelihood Model fitting Neuronal firing Neuronal networks Parameter estimation Point process Quadratic kernel Spike trains System identification Threshold element## Preview

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