Functional identification of the input-output transforms of mammalian motoneurones

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We studied the responses of rat hypoglossal and cat lumbar motoneurones to a variety of excitatory and inhibitory injected current transients during repetitive discharge. The amplitudes and time courses of the transients were comparable to those of the synaptic currents underlying postsynaptic potentials (PSPs) recorded in these cells. Poisson trains of these current transients were combined with an additional independent, high frequency random waveform to approximate band-limited white noise. The composite, white noise waveform was then superimposed on long duration suprathreshold current steps. We used the responses of the motoneurones to the white noise stimulus to derive zero-, first- and second-order Wiener kernels, which provide a quantitative description of the relation between injected current and discharge probability. The convolution integral computed for an injected current waveform and the first-order Wiener kernel provides the best linear prediction of the associated peristimulus time histogram (PSTH). This linear model provided good matches to most of the PSTHs compiled between the times of occurrence of individual current transients and motoneurone discharges. However, for the largest amplitude current transients, a significant improvement in the PSTH match was often achieved by expanding the model to include the convolution of the second-order Wiener kernel with the input. The overall transformation of current inputs into firing rate could be approximated by a second-order Wiener Model, i.e., a cascade of a dynamic, linear filter followed by a static non-linearity. At a given mean firing rate, the non-linear component of the motoneurone's response could be described by the square of the linear component multiplied by a constant coefficient. The amplitude of the response of the linear component increased with the avarage firing rate, whereas the value of the multiplicative coefficient in the nonlinear component decreased. As a result, the overall transform could be predicted from the mean firing rate and the linear impulse response, yielding a relatively simple, general description of the motoneurone's input-output function.