The maximum likelihood approach to the identification of neuronal firing systems

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_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 $$Prob\{ Y_t = 1|H_t \} = \Phi (U_t - \theta )$$ where $$U_t = \sum\limits_{u = 0}^{\gamma _t - 1} {a_u } X_{t - u} + \sum\limits_{u = 0}^{\gamma _t - 1} { \sum\limits_{\nu = 0}^{\gamma _t - 1} {b_{u,\nu } } X_{t - u} X_{t - \nu } } $$ where γ _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 $$U_t = \sum\limits_{u = 0}^{\gamma _t - 1} {a_u } X_{t - u} + \sum\limits_{u = 0}^{\gamma _t - 1} {b_u } X_{t - u} $$ with X_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 toAplysia 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.