Dynamical Systems in Neuroscience:

Simple Models 2958.2.1 Regular spiking (RS) neuronsRegular spiking neurons are the major class of excitatory neurons in the neocortex.Many are Class 1 excitable, as we show in Fig. 7.3 using in vitro recordings of a layer 5pyramidal cell of rat’s primary visual cortex; see also Tateno et al. (2004). RS neuronshave a transient K + current I A , whose slow inactivation delays the onset of the firstspike and increases the interspike period, and a persistent K + current I M , which isbelieved to be responsible for the spike frequency adaptation seen in Fig. 7.43. Let ususe the simple model (8.5, 8.6) to capture qualitative and some quantitative featuresof typical RS neurons.We assume that the resting membrane potential is v r = −60 mV and the instantaneousthreshold potential is v t = −40 mV; that is, instantaneous depolarizations above−40 mV cause the neuron to fire, as in Fig. 3.15. Assuming that the rheobase is 50pA, and the input resistance is 80 MΩ, we find k = 0.7 and b = −2. We take themembrane capacitance C = 100 pF, which yields a membrane time constant of 8 ms.Since b < 0, depolarizations of v decrease u as if the major slow current is theinactivating K + current I A . The inactivation time constant of I A is around 30 msin the subthreshold voltage range, hence we take a = 0.03 ≈ 1/30. The membranepotential of a typical RS neuron reaches the peak value v peak = +35 mV during aspike (the precise value has little effect on dynamics) and then repolarizes to c = −50mV or below, depending on the firing frequency. The parameter d describes the totalamount of outward minus inward currents activated during the spike and affecting theafter-spike behavior. Trying different values, we find that d = 100 gives a reasonableF-I relationship, at least in the low-frequency range.As it follows from Ex. 10, we can also interpret u as the membrane potential ofa passive dendritic compartment, taken with the minus sign. Thus, when b < 0, thevariable u represents the combined action of slow inactivation of I A and slow chargingof the dendritic tree. Both processes slow down the frequency of somatic spiking.Notice that we round-up all the parameters, e.g., we use d = 100 and not 93.27.Nevertheless, the simulated voltage responses in Fig. 8.12 agree quantitatively withthe in vitro recordings of the layer 5 pyramidal neuron used in Fig. 7.3. Tweakingthe parameters, considering multidimensional u, or adding multiple dendritic compartments,one can definitely improve the quantitative correspondence between the modeland the in vitro data of that particular neuron, but this is not our goal here. Instead,we want to understand the qualitative dynamics of RS neurons using the geometry oftheir phase portraits.Phase plane analysisFigure 8.13 shows recordings of two pyramidal RS neurons from the same slice whilean automated procedure injects pulses of dc-current to determine their rheobase. Theneuron on the left exhibits monotonically increasing (ramping) or decreasing responsesof membrane potential to weak input pulses, long latencies of the first spike and norebound spikes, whereas the neuron on the right exhibits non-monotone overshooting