Dynamical Systems in Neuroscience:

162 Conductance-Based Modelscurrent, I0-k(v-v r )(v-v t )+b(v-v r )I (V)-k(v-v r )(v-v t )I 0 (V)v rv tmembrane potential, vFigure 5.24: The relationship betweenthe parameters of the simple model(5.7, 5.8) and the instantaneous andsteady-state I-V relations, I 0 (V ) andI ∞ (V ), respectively.transforms the simple model into the equivalent form˙v = I + v 2 − u if v ≥ 1, then (5.5)˙u = a(bv − u) v ← c, u ← u + d (5.6)having only four dimensionless parameters.Derivation via I-V relationsThe parameters of the simple model can be derived using instantaneous (peak) andsteady-state I-V relations. Let us represent the model in the following equivalent formC ˙v = k(v − v r )(v − v t ) − u + I if v ≥ v peak , then (5.7)u = a{b(v − v r ) − u} v ← c, u ← u + d (5.8)where v is the membrane potential, u is the recovery current, and C is the membranecapacitance. The quadratic polynomial −k(v − v r )(v − v t ) approximates thesubthreshold part of the instantaneous I-V relation I 0 (V ). Here, v r is the resting membranepotential, and v t is the instantaneous threshold potential, as in Fig. 5.24. Thatis, instantaneous depolarizations above v t result in spike response. The polynomial−k(v − v r )(v − v t ) + b(v − v r ) approximates the subthreshold part of the steady-stateI-V relation I ∞ (V ). When b < 0, its maximum approximates the rheobase current ofthe neuron, i.e., the minimal amplitude of a dc-current needed to fire a cell. Its derivativewith respect to v at v = v r , i.e., b − k(v r − v t ), corresponds to the resting inputconductance, which is the inverse of the input resistance. Knowing both the rheobaseand the input resistance of a neuron, one could determine the parameters k and b, aswe do in Chap. 8. This method does not work when b > 0.The sum of all slow currents that modulate the spike-generation mechanism arecombined in the phenomenological variable u with outward currents taken with theplus sign. The form of (5.8) ensures that u = 0 at rest, i.e., when I = 0 and v = v r .The sign of b determines whether u is an amplifying (b < 0) or a resonant (b > 0)variable. In the latter case, the neuron sags in response to hyperpolarized pulses ofcurrent, peaks in response to depolarized subthreshold pulses, and produces rebound(post-inhibitory) responses. The recovery time constant is a. The spike cut-off value is