Class-II neurons

The situation may be different for II neurons. Recall that the latencies of class-II neurons are approximately independent of input intensity (Fig. 2.10). Therefore, the phase equation for the super-threshold regime only has to supply this fixed delay. Since this latency is short, the role of the super-threshold equation in initiating a spike might be negligible. However, the super-threshold equation also handles the absolute and relative refractory period after a spike. These are important, but the dynamics during the refractory period is very stereotyped.

In their spike-response model, Kistler et al. (1997) convolve the input with a kernel, which may depend on the time of the previous spike, to calculate the membrane potential.

If this potential reaches a threshold a spike is simulated by a stereotyped voltage trace which is added on the voltage trace. The performance of the spike-response model to predict spikes of the Hodgkin-Huxley model indeed is good (Kistler et al., 1997). Whether this one-dimensional nonlinear model is able to reproduce the super-threshold behavior of a class-II neuron was not shown. With the stimuli they used to test their model, mainly the sub-threshold dynamics of the model was tested.

3.8 Summary

The neuron’s f -I-curve determines completely a non-leaky phase oscillator ˙ϕ f I t , which predicts the timing of spikes well for currents fluctuations slower than an interspike interval.

A spiking neuron acts like a low-pass filter with its cut-off frequency determined by the actual firing frequency.

Exploring the dynamics on limit cycles in conductance-based models leads to the leaky phase oscillator, which gives a new interpretation of the integrate-&-fire neu-ron.

Analyzing the properties of the saddle-node bifurcation of class-I neurons results in theθ-model as proposed by Ermentrout (1996).

Latencies measured in class-I conductance-based models contradict the latencies predicted by phase oscillators.

The dynamics of the sub-threshold regime cannot be extrapolated from the dynam-ics on limit cycles of the super-threshold regime.

Two-dimensional models are needed to combine the sub-threshold dynamics with the dynamics on limit cycles for class-I neurons.

4 Dynamics of Spike-Frequency Adaptation

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Contents

4.1 From channels to firing frequency . . . 65 4.1.1 Encoder adaptation . . . 66 M-type currents . . . 70 AHP-currents . . . 73 Slow recovery from inactivation . . . 74 General model for encoder adaptation . . . 75 Physical stimulus as input . . . 75 4.1.2 Transducer adaptation . . . 77 4.1.3 Depressing synapses . . . 78 4.2 Defining the parameters of the adaptation models . . . 78 4.2.1 Steady-state strength of adaptation . . . 79 Encoder adaptation . . . 79 Transducer adaptation . . . 79 4.2.2 Theγ f - term . . . 80 4.2.3 Time constants of adaptation . . . 80 4.3 Signal transmission properties . . . 83 4.3.1 Adapted f -I-curves . . . . 83 4.3.2 Linear adaptation and linear steady-state f -I-curves . . . . . 85 4.3.3 High-pass filter properties due to adaptation . . . 88 4.3.4 Time-derivative detection . . . 90 4.3.5 Nonlinear effects . . . 90 4.4 Combination with spike-generator . . . 93 4.4.1 Adaptation and spike dynamics . . . 94 4.5 Discussion . . . 94 4.5.1 Models of spike-frequency adaptation . . . 94 4.5.2 Assumptions of the model for encoder adaptation . . . 95 4.5.3 Biophysical mechanisms of spike-frequency adaptation . . . 97 4.5.4 Transducer adaptation and depressing synapses . . . 99 4.5.5 Functional role of adaptation . . . 99 4.6 Summary . . . 100

Cover: Voltage trace recorded in an adapting auditory receptor of Locusta migratoria. Overlaid is the time course of the firing frequency and a fit of the adaptation model.

Spike-frequency adaptation is a widespread phenomenon in spiking neurons, exhib-ited by almost any type of neuron in vertebrates as well as in invertebrates, in peripheral

63

Firing Frequency

Input Spikes: Rate r(t) Depressing Synapse

divisive

Firing Frequency

J/A Physical Stimulus J

Transducer Adaptation

subtractive

divisive

Firing Frequency Physical Stimulus

Dendritic Input I

Input Current

- AHP-currents - M-type currents Encoder Adaptation

- Slow recovery from inactivation subtractive

I I

f(t)

r (1-A)

f(t) J-A

g(J/A) g(J-A)

f(t) J

I

I-A g(J)

PSfrag replacements A

B

C

Figure 4.1: DIFFERENT MECHANISMS OF SPIKE-FREQUENCY ADAPTATION. AVarious types of slow ionic currents (M-type, AHP, recovery from inactivation) act subtractive on the input current I. They cause spike-frequency adaptation in dependence on the output firing frequency ft of the spike generator. This type of adaptation is referred to as “encoder adaptation” in this thesis. The input current I is either injected by a microelectrode, or is the current from the dendritic tree, or flows through a conductance gJ, which is changed by the transduction of a physical stimulus J. B In receptor neurons the transduction process of a physical stimulus J into a change of a conductance gJ may adapt (transducer adaptation). This type of adaptation depends on the input intensity J. It acts on the input J either subtractive or divisive. CA continous stimulation of a synapse with a spike train may gradually reduce its efficiency. Such depressing synapses act divisive on the input rate rt . The postsynaptic current is reduced, and the resulting output spikes show spike-frequency adaptation.

as well as in central neurons. For example, it was observed in rat pyramidal neurons (Gustafsson & Wigstr¨om, 1981; Lanthorn et al., 1984; Barkai & Hasselmo, 1994), rat motoneurons (Granit et al., 1963), cat layer V cells of the sensorimotor cortex (Stafstrom et al., 1984), pyramidal tract cells of the cat (Koike et al., 1970), cat motoneurons (Ker-nell, 1965), cat auditory-nerve fiber (Javel, 1996), arthropod stretch receptors (Edman et al., 1987a; French, 1989b), auditory receptors in noctuid moths (Coro et al., 1998) and locusts (Michelsen, 1966).

Adapting neurons respond to a prolonged constant stimulation with a decaying firing frequency. The final firing frequency levels out at some value below the initial one, or the neuron may even stop spiking after a while. The time scales associated with spike-frequency adaptation range from tens of milliseconds (Madison & Nicoll, 1984; Schwindt et al., 1988; Storm, 1989; Engel et al., 1999; Sah & Clements, 1999; Stocker et al., 1999) to several seconds (Edman et al., 1987a; French, 1989b; Hocherman et al., 1992).

There are many different mechanism causing spike-frequency adaptation. Focusing on single cell properties and neglecting network effects like recurrent inhibition, these mechanisms can be summarized in three basic groups (see Fig. 4.1).

First, there exist slow ionic currents, which directly influence the initiation of spikes in dependence on the generated firing frequency. For this adaptation mechanism the term

“encoder adaptation” is used throughout this thesis. Three main groups of such ionic cur-rents can be distinguished: M-type curcur-rents, which are caused by slow, voltage-dependent, high-threshold potassium channels (Brown & Adams, 1980; Halliwell & Adams, 1982), AHP-currents mediated by calcium-dependent potassium channels (Madison & Nicoll, 1984), and slow recovery from inactivation, e.g. of the fast sodium channel (Fleidervish et al., 1996). Properties of these and other ionic currents are presented in detail in the discussion.

Second, the transduction process of a physical stimulus into a change of a membrane conductance may adapt itself in dependence on the stimulus intensity. This type of adap-tation is important for receptor neurons. It is referred to as “transducer adapadap-tation”.

Third, neurons usually receive their input via synapses, which may adapt as well.

Such synapses are known as depressing synapses.

Various modeling studies on spike-frequency adaptation have already been performed, most of them based on conductance based models of a specific cell, thus focusing on a special encoder-adaptation mechanism (Cartling, 1996a; Wang, 1998; Ermentrout, 1998).

In this chapter an attempt is made to formulate a general model for different biophysical mechanisms inducing encoder adaptation. Analyzing the effects of various slow ionic cur-rents on the neuron’s intensity-response curve ( f -I-curve), a universal phenomenological model for encoder adaptation is derived. Its parameters can be easily accessed experi-mentally by measuring the neuron’s f -I-curves. Comparable models are derived for some types of transducer adaptation as well as for depressing synapses.

Based on the model, signal transmission properties due to spike-frequency adaptation can be quantified from the neuron’s f -I-curves. For encoder adaptation and some types of transducer adaptation spike-frequency adaptation acts subtractive on the input (section 4.3.1) and turns the neuron into a high-pass filter (section 4.3.3). Depending on the adap-tation time-constant and the shape of the neuron’s f -I-curves, the response of the neuron resembles the time-derivative of the stimulus (section 4.3.4). Together with the inherent threshold-property of the spike-generator, adaptation is more than just filtering out the mean intensity of the stimulus. It also leads to effects like suppression of