Dynamical Systems in Neuroscience:

Simple Models 301Fig. 8.17a, show a reasonable fit. The simple model predicts more than 90% of spikesof the in vitro neuron, often with a submillisecond precision, see Fig. 8.17c. Of course,we should not expect to get a total fit, since we do not explicitly model the sourcesof intrinsic and synaptic noise present in the cortical slice. In fact, presentation of thesame input to the same neuron a few minutes later produces a response with spikejitter, missing spikes, and extra spikes (as in Fig. 7.24) comparable with those in thesimulated response.8.2.2 Intrinsically bursting (IB) neuronsThe class of intrinsically bursting (IB) neurons forms a continuum of cells that differin their degree of “burstiness”, and it probably should consists of subclasses. On oneextreme, responses of IB neurons to injected pulses of dc-current have initial stereotypicalbursts (Fig. 8.18a) of high-frequency spikes followed by low-frequency tonicspiking. Many IB neurons bursts even when the current is barely superthreshold andnot strong enough to elicit a sustained response (as in Fig. 8.21, bottom). On theother extreme, bursts could be seen only in response to sufficiently strong current,as in Fig. 8.11 or Fig. 9.1b. Weaker stimulation elicits regular spiking responses. Incomparison with typical RS neurons, the regular spiking response of IB neurons haslower firing frequency, higher rheobase (threshold) current, exhibit shorter latency tothe first spike and noticeable afterdepolarizations (ADPs), compare RS and IB cell inFig. 8.11.Magnifications of the responses of two IB neurons in Fig. 8.18b and c show that theinterspike intervals within the burst may be increasing or decreasing, reflecting possiblydifferent ionic mechanisms of burst generation and termination. In any case, the initialhigh-frequency spiking is caused by the excess of the inward current or the deficit ofthe outward current needed to repolarize the membrane potential below the threshold.As a result, many spikes are needed to build-up outward current to terminate thehigh-frequency burst. After the neuron recovers, it fires low frequency tonic spikesbecause there is a residual outward current (or residual inactivation of inward current)that prevents the occurrence of another burst. Many IB neurons can actually fire twoor more bursts before they switch into tonic spiking mode, as in Fig. 8.18a. Belowwe present two models of IB neurons, one relying on the interplay of voltage-gatedcurrents, another relying on the interplay of fast somatic and slow dendritic spikes.Let us use the available data on the IB neuron in Fig. 8.11 to build a simple onecompartmentmodel (8.5, 8.6) exhibiting IB firing patterns. The neuron has restingstate at v r = −75 mV and instantaneous threshold at v t = −45 mV. Its rheobase is350 pA, and the input resistance is around 30 MΩ, resulting in k = 1.2 and b = 5.The peak of the spike is at +50 mV, and the after-spike resetting point is aroundc = −56 mV. The parameters a = 0.01 and d = 130 give a reasonable fit of theneuron’s current-frequency relationship.The phase portraits in Fig. 8.19 explain the mechanism of firing of IB patterns inthe simple model. When I = 0, the model has an equilibrium at −75 mV, which is