T i represents tonic input currents of vestibular origin, Bi(t)Bi(t) represents saccadic burst command inputs, and InoiseInoise is a noise current. WijsjWijsj gives the recurrent input from neuron j to i , where W ij is the connection strength and sj(uj,t)sj(uj,t) is the synaptic activation. The synaptic activation functions sj(uj,t)sj(uj,t) are governed by a two time-constant approach (Supplemental Methods) to steady-state
activation functions s∞,j(rj)s∞,j(rj). s∞,j(r)s∞,j(r) were chosen from a two-parameter family of functions that increase from 0 at r = 0 to 1 at large r: equation(Equation 3) s∞,j(r)=b∞,j11+exp(Rf,j−r)/Θj−a∞,j,wherea∞,j=11+expRf,j/Θj,b∞,j=11−a∞,j. Rf,jRf,j gives the inflection point: s∞,j(r)s∞,j(r) is superlinear for r
narrow range of r for small ΘjΘj and increases gently for large ΘjΘj. This family allowed us to generate a wide range of sigmoidal, saturating, and approximately linear curves within the relevant range of r . Synaptic activation curves s∞,j(rj)s∞,j(rj) were chosen to be different for excitatory and inhibitory synapses, but the same within each synapse type. The model fitting procedure was conducted in two steps. First, we fit a conductance-based model neuron that reproduced the current injection experiments of Figure 2D. Second, we incorporated this conductance-based neuron into a circuit model of the goldfish oculomotor integrator and used a constrained regression procedure to fit the connectivity parameters W ij and T i of the circuit model for different Montelukast Sodium choices of the steady-state synaptic activation functions s∞(r)s∞(r). Single-Neuron Model Calibration. Parameters of the intrinsic ionic conductances were calibrated to accurately match the current injection experiments illustrated in Figure 2D. In the experiments, slow up-and-down ramps of injected current drove the recorded neuron
across the firing-rate range observed during fixations. The model neuron’s parameters were optimized to reduce the least-squares distance between the experimental and simulated cumulative sum of the spike train as a function of time ( Figure 3B). Parameter optimization was performed using the Nelder-Mead downhill simplex algorithm. To obtain the steady-state response curve r=f(I)r=f(I) (Figure 3C), the single-neuron model was injected for 60 s with constant currents of various, finely discretized strengths, and the firing rate r was found from the inverse interspike intervals, discarding the first 5 s to assure convergence to steady-state. A noise current Iinoise(t) was included to approximately match the coefficient of variation of interspike intervals observed experimentally ( Aksay et al., 2003; Supplemental Methods). Fitting the Recurrent Connectivity.