Dynamical Systems in Neuroscience:

Excitability 265modelI K(M)I Na,p +I K -modelC ˙V{ }} { { }} {= I −g L (V −E L )−g Na m ∞ (V )(V −E Na )−g K n(V −E K ) − g M n M (V −E K )ṅ = (n ∞ (V ) − n)/τ(V )ṅ M = (n ∞,M (V ) − n M )/τ M (V ) (slow K + M-current)(7.1)whose excitable and spiking properties are similar to those of the I Na,p +I K -submodelon a short time scale. However, the long-term behavior of the two models may be quitedifferent. For example, the K + M-current may result in frequency adaptation during along train of action potentials. It can change the shape of the I-V relation of the modeland result in slow oscillations, post-inhibitory spikes, and other resonator propertieseven when the I Na,p +I K -submodel is an integrator. All these interesting phenomenaare discussed in this section.In general, models having fast and slow currents, such as (7.1), can be written inthe fast-slow formẋ = f(x, u) (fast spiking),˙u = µg(x, u) (slow modulation),(7.2)where the vector x ∈ R m describes fast variables responsible for spiking. It includes themembrane potential V , activation and inactivation gating variables for fast currents,etc. The vector u ∈ R k describes relatively slow variables that modulate fast spiking,e.g., the gating variable of a slow K + current, the intracellular concentration of Ca 2+ions, etc. The small parameter µ represents the ratio of time scales between spikingand its modulation. Such systems often result in bursting activity, and we study themin detail in Chap. 9.7.3.1 Spike-frequency modulationSlow currents can modulate the instantaneous spiking frequency of a long train ofaction potentials, as we illustrate in Fig. 7.43a using recordings of a layer 5 pyramidalneuron. The neuron generates a train of spikes with increasing interspike interval (seeinset in the figure) in response to a long pulse of injected dc-current. In Fig. 7.43bwe plot the instantaneous interspike intervals T i , i.e., the time intervals between spikesnumber i and i + 1, as a function of the magnitude of injected current I. Notice thatT i (I) < T i+1 (I), meaning that the intervals increase with each spike. The functionT 0 (I) describes the latency of the first spike, and T ∞ (I) describes the steady-state(asymptotic) interspike period. The instantaneous frequencies are defined as F i (I) =1000/T i (I) (Hz), and they are depicted in Fig. 7.43c. Since the neuron is Class 1excitable, the F-I curves are square-root parabolas (see Sect. 6.1.2). Notice that F 0 (I)is nearly a straight line, probably reflecting the passive charging of the dendritic tree.Decrease of the instantaneous spiking frequency, as in Fig. 7.43, is referred to asspike-frequency adaptation. This is a prominent feature of cortical pyramidal neuronsof the regular spiking (RS) type (Connors and Gutnick 1990), as well as many