Spike-Frequency Adaptation Jan Benda and Joel Tabak Encyclopedia of Computational Neuroscience, Springer 2014
Definition When stimulated with a constant stimulus, many neurons initially respond with a high spike frequency that then decays down to a lower steady-state frequency (Fig. 1 A). This dynamics of the spike frequency response is referred to as “spike-frequency adaptation”. Spike-frequency adaptation is a process that is slower than the dynamics of action potential generation. Spikefrequency adaptation by this definition is an aspect of the neuron’s super-threshold firing regime, although the mechanisms causing spike-frequency adaptation could also be at work in the neuron’s subthreshold regime.
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Figure 1: Phenomenon spike-frequency adaptation. A Spike train (vertical strokes at the top) of an adapting neuron evoked by the onset of a constant stimulus (green bar at the bottom, I = 15 nS). The spike frequency drops from an initially high onset rate, f0 , in an approximately exponential way down to a lower steady state rate, f∞ . B Repeating the experiment shown in A for various stimulus intensities I (here an injected current) results in the onset f -I curve f0 (I) and the steady-state f -I curve f∞ (I). Shown are simulations of a single-compartment conductance-based model with an M-type current (the Ermentrout model, Ermentrout, 1998; Benda et al., 2010, with g¯Ca = 0, g¯AHP = 0, and g¯M = 8 µS/cm2 , the input current is scaled such that 1 nA equals 1 µA/cm2 ).
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Detailed Description In the context of spike-frequency adaptation, spike frequency is often measured as the instantaneous rate, i.e., the averaged reciprocal interspike-intervals at each time. This measures the inverse period of super-threshold firing. In contrast, the PSTH (peristimulus time histogram) estimates the probability of a spike at a given time and therefore is also sensitive to variability of the response caused by intrinsic or external noise sources. Important characteristics of a neuron with spike-frequency adaptation are its f -I curves (spike frequency f versus input strength I). Because of adaptation, a certain input intensity will not evoke a constant spike frequency. Instead, the initial response in spike frequency to the onset of a constant input will result in the so-called “onset f -I curve” (green curve in Fig. 1 B). Later during the constant stimuli, the spike frequency reaches a steady state. Drawn as a function of input intensity, this forms the “steady-state f -I curve”, which usually lies below the onset f -I curve, because the spike frequency adapts from the onset down to the steady state (red curve in Fig. 1 B). The third type of f -I curves is the “adapted f -I curves” (Benda and Herz, 2003). For these the neuron is first adapted to some adaptation input of fixed intensity and then the onset responses to various test inputs are measured (Fig. 2). The adapted response curves indicate how the neuron being in a certain and fixed adaptation state will initially respond to new stimuli. While each neuron has exactly one onset and one steady-state f -I curve, it has for each adaptation stimulus a different adapted f -I curve. 250
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Figure 2: Adapted f -I curves. In addition to the onset and the steady-state f -I curves, the adapted f -I curves provide important information about the adaptation properties of a neuron. A In order to measure an adapted f -I curve for a specific adaptation state, the neuron is first adapted to its steadystate response f∞ (I0 ) by stimulating it with a constant adapting stimulus I0 (here 15 nA). Then, a test stimulus I is applied and the onset response f0 (I, I0 ) to this stimulus is measured. In the figure, the responses (red and orange) to two different test stimuli (I = 20 nAin green and I = 10 nAin dark cyan) are shown. B Each adapting stimulus I0 results in one adapted f -I curve f0 (I, I0 ). Shown are adapted f -I curves for I0 = 5, 10, 15, 20 nA. In this example the adaptation process mainly shifts the adapted f -I curves to the right. All curves together characterize the spike-frequency response of an adapting neuron. Same model neuron as in Fig. 1.
Simple adaptation currents (see below) will shift the neuron’s adapted f -I curves to higher input intensities. Other (less well-understood) mechanisms like presynaptic inhibition may also
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tilt the adapted f -I curves (Hildebrandt et al., 2011) or distort it in other ways (see also Fig. 5). Spike-frequency adaptation is related to the classification of neurons into phasic, phasictonic, or tonic spiking neurons. Tonic spiking neurons are neurons that keep firing in response to a constant stimulus with an almost constant rate. Many neurons that are classified as tonic spiking, however, still show some weak spike-frequency adaptation. On the other hand, phasic spiking neurons only emit one or several spikes to the onset of the stimulus and then cease spiking. If this onset response consists of several spikes with a decreasing spike frequency, then the phasic response is probably caused by strong spike-frequency adaptation. With less or only a single spike and no clear decrease in spike frequency, this type of response can be that of a type III neuron. The intermediate phasic-tonic responses exactly match the definition of spike-frequency adaptation.
Mechanisms There are many different mechanisms that all cause a neuron to adapt its spike-frequency response. First, there are adaptation currents — ionic currents that act together with the spikegenerating currents (“output-driven adaptation”). Second, other mechanisms, such as fatigue of receptor currents, adapt the input a neuron receives (“input-driven adaptation”). And third, there are network effects that also can adapt the neuron’s response in both an input-driven or output-driven way. Adaptation Currents
There exists a whole zoo of mechanisms based on ionic currents that are all directly or indirectly activated by the action potentials generated by the neuron — the output of the neuron — and have at least one time scale involved that is slower than the dynamics of an action potential. M-type currents are a simple example (Fig. 3, Brown and Adams, 1980). These are voltagegated potassium currents that are mainly activated at high voltages as they occur during an action potential and that deactivate with a slow time constant of about 100 ms. If the neuron initially fires with a frequency of say 200 Hz, then the activation of the M-type currents builds up from one action potential to the next, since there is not enough time for the current to completely deactivate between succeeding action potentials. As a potassium current, the activated M-type currents then counteract the input to the neuron and as a consequence the spike frequency is reduced — the neural response adapts. An important class of adaptation currents are AHP-type currents, calcium-gated potassium currents, that enhance the afterhyperpolarization following an action potential (Sah, 1996). Here, during each action potential, calcium enters the cell through voltage-gated calcium channels. Depending on the intracellular calcium concentration, the AHP-type currents are activated and as potassium currents inhibit the input to the neuron. Calcium is only slowly removed from the cytosol and this time scale is then reflected by the resulting spike-frequency adaptation. Further ionic mechanisms that cause spike-frequency adaptation include sodium-activated potassium channels and slow inactivation of the sodium current (for details see “Ion Channels and Pumps in Neuronal Adaptation”).
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Figure 3: Dynamics of an adaptation current. A The membrane potential V in response to a step input (I = 15 nA, indicated by the green bar) simulated with the same model as in Fig. 1. As adaptation progresses the interspike intervals get longer. B The gating variable of the M-type current a is increased by every action potential and decays slowly in between them. This way a slowly builds up during stimulation. C The adaptation current, here an M-type current, IM , gets larger and larger as a increases and inhibits the input. This way it causes the neuron’s response to adapt.
Input Adaptation
In receptor neurons the whole transduction process from the physical stimulus to the receptor current can adapt in different ways. These can be mechanisms controlling the sensitivity of the receptor organ, like the pupillary light reflex. Within the receptor neuron itself, the transduction machinery may produce adapting receptor currents due to fatigue (e.g., bleaching of rhodopsin) or active adaptation to the intensity of the physical stimulus within a second messenger cascade (e.g., in photoreceptors) or by molecular motors (e.g., in mechanical transduction in hair cells). All these and many other mechanisms eventually produce an adapting receptor current that then leads to spike-frequency adaptation in the receptor neuron. Gollisch and Herz (2004) demonstrated how to unmask input-driven adaptation from output-driven adaptation in auditory receptor neurons of locusts. In all other neurons that receive their input through synapses on their dendrite, the input current reaching the site of action-potential generation can also be adapted. The various forms of synaptic dynamics, in particular short-term depression, are potential sources of spike-frequency adaptation. Also various ionic currents on the dendrite might potentially adapt the input.
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Network Effects
In case the spiking activity of a neuron leads to the activation of recurrent inhibitory synapses, this might cause spike-frequency adaptation in a way quite similar as the adaptation currents (Sutherland et al., 2009). Here, the slow time scale would be introduced by the time scale of the postsynaptic current of the inhibitory synapse, by the delay introduced by wiring, and by the integration dynamics of the neurons participating in this negative feedback loop. Similar to input-driven adaptation, some forms of feedforward input may also cause spikefrequency adaptation in the target neuron. In its simplest form, the presynaptic neuron already adapts, and thus the postsynaptic neuron very likely will adapt as well. In addition, the synaptic transmission might adapt through synaptic plasticity or by presynaptic inhibition, or delayed and slower inhibitory input can adapt the response of the target neuron. Network and cell-intrinsic adaptation mechanisms shape in specific ways functionally different response properties of auditory interneurons in locusts (Hildebrandt et al., 2009).
Models of Spike-Frequency Adaptation There are many ways to model spike-frequency adaptation. In the following overview, we focus on output-driven adaptation, i.e., adaptation currents that are activated by the neuron’s output activity (membrane voltage with action potentials). In contrast, input-driven adaptation is activated by the input signal irrespective of the resulting output. Conductance-Based Models
In conductance-based models like the Hodgkin-Huxley model, any ionic current of a neuron is modeled in detail. This requires exact knowledge of the specific type of adaptation current involved. For example, the M-type current IM would be modeled as IM = g¯M a(V − EK ) da τa (V ) = a∞ (V ) − a dt
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where EK is the potassium reversal potential, g¯M is the maximum conductance, and a is the gating variable of this voltage-gated current. τa (V ) and a∞ (V ) are the voltage-dependent time constant and activation variable, respectively, that determine the (slow) temporal evolution of the gating variable. a∞ (V ) is a sigmoidal function depending on the membrane potential V that is close to zero at the resting potential and about one during an action potential (Brown and Adams, 1980; Benda and Herz, 2003). Integrate-and-Fire Models
As for conductance-based models, any adaptation current can also be simply added to the input of an integrate-and-fire model. The only difference is that the effect the action potential has on the dynamics of the current has to be modeled explicitly, for example, by increasing a in equation (2) by a certain increment whenever there was a spike. Often, however, a more generalized form of adaptation currents is used in (leaky, quadratic, exponential) integrate-and-fire neurons (Izhikevich, 2003; Brette and Gerstner, 2005). This has
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the advantage that the exact nature of the adaptation current in question does not have to be known. First, the electromotoric force EK − V is often approximated away, so that the gating variable a then becomes the (appropriately scaled) adaptation current A. Second, a∞ (V ) is replaced by incrementing A by some appropriate value ∆A whenever an action potential occurs at time ti . Between the action potentials A decays exponentially with a fixed adaptation timeconstant τa : IM ≈ A ∞ dA = −A + ∆A ∑ δ(t − ti ) . τa dt i=−∞
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The adaptation dynamics (4) can be extended by an additional term proportional to V in order to incorporate the fact that some adaptation currents are already activated by subthreshold membrane voltages. Integrate-and-fire models with a dynamic threshold are another class of adapting neuron models, where the firing threshold is incremented by each action potential and follows a slow dynamics between the action potentials. Although this mechanism also produces spike-frequency adaptation (Liu and Wang, 2001), the resulting adapted f -I curves do not only shift to higher inputs, as any adaptation current does, but also reduce their slopes (Benda et al., 2010). Firing-Rate Models
By simply approximating the neuron’s spike generator by its f -I curve f0 (I) and using Kirchhoff’s law that ionic currents over the cell’s membrane add up yields the following equation for the spike frequency f (t): f = f0 (I − A) dA = −A + A∞ ( f ) τa dt
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(Benda and Herz, 2003). Because the adaptation current acts inhibitory on the input current, it is subtracted from the input. The dynamics for A is the same as for the simplified integrateand-fire neurons, but it is driven via the term A∞ ( f ) by the output spike frequency and not by individual spikes any more. This dependence on the spike frequency arises from averaging over the spike train in time (Wang, 1998). This averaging procedure is possible if the spike frequency is higher than the reciprocal adaptation time-constant. Only then the adaptation dynamics can be separated from the one of the membrane potential. Choosing A∞ ( f ) to be proportional to f often is a reasonable approximation. Then, the only nonlinearity remaining is the neuron’s onset f -I curve. By linearizing the onset f -I curve around the current spike frequency, one can easily compute the neuron’s filter properties that are introduced by the adaptation process.
Signal Processing The slow adaptation variable (gating variable of M-type current, intracellular calcium concentration for AHP-type currents, etc.) is a low-pass filtered version of the output spike train (output-driven adaptation, Eqs. (4) or (6)) or of the input to the neuron (in the case of inputdriven adaptation). Since this low-pass filtered signal is then subtracted from the input current
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Figure 4: High-pass filter of adapting neurons. Adaptation processes add high-pass filter characteristics to a neuron’s transfer function between the input to the neuron (here, e.g., an injected current) and the resulting spike-frequency. Shown is the gain, i.e., the amplitude of the modulation of the spike-frequency response divided by the amplitude of the injected current input as a function of the frequency of that input, obtained for the same model neuron as in Fig. 1. As a stimulus a low-pass filtered Gaussian white noise with cutoff frequency at 16 Hz, mean 10 nA, and standard deviation 2 nA was used. The slopes of the steady-state f -I curve (red) at 10 nA and the one of the adapted f -I curve (green) for I0 = 10 nA are plotted for comparison.
I, this basically (i.e., with linear f -I curves) results in a high-pass filter between the input to the neuron and the resulting output spike frequency. The spike frequency of an adapting neuron thus encodes the high-pass filtered input signal (Fig. 4; Benda and Herz, 2003). For example, in electroreceptors, this high-pass filter resulting from adaptation processes enhances the response to fast stimulus components in communication signals (Benda et al., 2005). The gain of the filter at low stimulus frequencies that are below the reciprocal adaptation time-constant is given by the slope of the steady-state f -I curve. The gain of stimulus frequencies faster than the reciprocal adaptation time-constant is given by the slope of the adapted f -I curves. In the case of purely subtractive adaptation the latter equals the slope of the onset f -I curve. It follows that neurons with flat steady-state f -I curves (zero slope) implement perfect intensity invariance (zero gain at low stimulus frequencies), i.e., no matter on what mean intensity a certain input waveform is delivered, the output spike-frequency will always be the same, since the mean intensity is completely filtered out by the adaptation process (Benda and Hennig, 2008). The high-pass filter effect of spike-frequency adaptation considers the linear aspects of spike-frequency adaptation only. At least the steady-state f -I curve is indeed linearized by adaptation currents (as shown in Ermentrout, 1998, and also illustrated in Fig. 1 B). In addition, the non-linear shape of a neuron’s f -I curves (e.g., rectification at zero firing rate and saturation) together with the high-pass filter introduces additional aspects. For example, the subtractive shift of the adapted f -I curves to higher input intensities might be accompanied by a reduction in their maximum firing rate and/or by a reduction of their slope (see Fig. 5 for simple examples) and thereby influence the filter properties of a neuron in a non-linear way.
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The various forms of this interplay of adaptation dynamics and the resulting high-pass characteristics with nonlinearities lead to diverse functional consequences. For example, in the auditory system of crickets, the slow calcium dynamics leads to forward masking of weaker stimuli (Sobel and Tank, 1994). A collision-detecting neuron in the visual system of locusts responds very selectively to looming stimuli because of spike-frequency adaptation (Peron and Gabbiani, 2009). 300
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Figure 5: Nonlinearities and adaptation. Simple subtractive adaptation can have unexpected effects when it occurs after a nonlinearity. The examples show two simple scenarios where the firing rate f of a neuron is directly proportional to the input current J , and an adaptation current A acts subtractive on the current, as in Eq. (5): f = c · (J − A), with c > 0, and f = 0 if J < A. The input current, however, depends non-linearly on some (sensory) input I : J(I). A The input nonlinearity is a sigmoidal function of the sensory input: J(I) = (1 + exp(−(J − 6)/4))−1 − 0.2 (c = 320 Hz). This is the case, for example, in auditory or olfactory receptor neurons. Although the adaptation acts subtractive on the input current, the resulting adapted f -I curves appear to be shifted to the right as well as downwards. B J(I) = cos(0.2J) + 0.2 (c = 180 Hz). This models, for example, an orientation-selective cell in the visual system. Here, the adapted f -I curves appear to be shifted downward.
Adaptation currents can even alter the type of action-potential dynamics. A type I neuron where repetitive firing occurs via a saddle-node bifurcation on an invariant cycle can switch into a type II neuron where the resting potential loses stability via a Hopf bifurcation by the activation of an M-type adaptation current (Ermentrout et al., 2001; Prescott et al., 2006).
Multiple Adaptation Time-Scales Usually, spike-frequency adaptation occurs not only on a single but on several time scales. The shortest time scales of spike-frequency adaptation are in the range of 10 to 100 ms. This at the same time usually also is the strongest adaptation process in the neuron. Additional slower adaptation processes are weaker and have time constants of seconds, 10 seconds and slower. Adaptation on several time scales could resemble power-law adaptation (Clarke et al., 2013) or logarithmic adaptation (Xu et al., 1996) in contrast to the single adaptation process showing exponential adaptation of the firing rate.
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Noise Adaptation currents are a potential source of ion channel noise. Since these ionic currents are characterized by their slow adaptation dynamics, the resulting channel noise is a colored noise with correlation time given by the adaptation time-constant. Such a noise results in positive correlations between successive interspike intervals and in interspike-interval histograms with a sharper peak and heavier tail in comparison to the ones obtained with white noise (Fig. 6 D–F; Schwalger et al., 2010). Output-driven adaptation also interacts with white noise, e.g., generated by ion channel noise from fast spike-generating currents. In this case, adaptation introduces negative serial correlations between successive interspike intervals (Fig. 6 A–C; Chacron et al., 2000; Benda et al., 2010) that improve information transmission at low frequencies (Chacron et al., 2001). The interspike-interval statistics of auditory receptor neurons in locusts have been shown to be shaped by both white noise sources and colored noise presumably generated by an adaptation current (Fisch et al., 2012).
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Figure 6: Spike-frequency adaptation and noise. A–C Interspike -interval (ISI ) histogram (A), return map B, and ISI correlations C for the conductance-based model with an M-type adaptation current (same model as in Fig. 1) driven with white noise (mean I = 10 nA, noise strength D = 1 nA2 /Hz). D–F The same measures for colored noise arising from the stochasticity of the adaptation current. Noise strength was scaled to result in the same standard deviation of the ISI s (D = 30 nA2 /Hz). Note the more peaked shape of the ISI distribution as well as the positive ISI correlations.
Rhythmogenesis in Neural Networks The slow decrease in firing frequency due to adaptation currents provides delayed negative feedback that can result in the production of oscillations by excitatory neural networks. The population activity of an excitatory neural network can be roughly described by its mean firing
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rate, which varies according to df = f0 (I) − f (7) dt (Wilson and Cowan, 1972). Here, the firing frequency f is averaged spatially (over the population) and temporally (over the synaptic integration time scale). It is assumed that the neurons fire asynchronously. f0 is the steady-state firing rate that will be reached with a network integration time-constant τ. It is similar to the function f0 in Eq. (5), but is not strictly the firing rate of one neuron, as it also incorporates information about synaptic dynamics, heterogeneity in the neuron population, etc. Because of the recurrent excitatory connectivity, the input I to the network is given by I = w f , where f is the output firing rate of the network, and w is a connectivity parameter akin to synaptic weight or the number of connections per neuron. If neurons are adapting, the effective input becomes I = w f − A, where A is the average adaptation current in the population. This description is valid as long as adaptation is much slower than network integration (τa > τ). τ
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Figure 7: Oscillatory activity in an excitatory network with adaptation. A Time courses of f , the average firing rate (black), and A, the average adaptation current (blue) in the population. During an episode of activity ( f ' 1), A is increasing, until activity drops to ' 0. During the inter-episode interval, A decreases until the network is excitable enough to start a new high-activity episode. B The two time courses from A describe a closed trajectory in the (A, f ) plane. The red curve is the curve of steadystates of f , solutions of Eq. (8) at each value of A. There is an interval of A over which there are two stable steady states for f , one high and one low (and an intermediate, unstable steady state, dashed). The blue curve represents A∞ ( f ); when the phase point (A, f ) is above this curve A increases, below this curve A decreases, according to Eq. (6). During an episode of network activity (a), f is high so A increases, moving the trajectory to the right. After the point where the high steady state meets the unstable steady state, the only available steady state is at low f . The trajectory jumps down (b), starting the inter-episode interval (c), during which A decreases. When A is low enough, the lower steady state disappears and the trajectory jumps up (d), beginning a new high activity episode.
The population firing rate equilibrates to f = f0 (w f − A)
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and A varies according to Eq. (6). Note that Eq. (8) can have two stable solutions over a range of A values. This bistability, due to the fast positive feedback provided by excitatory connections, together with the slow A dynamics, produce relaxation oscillations as illustrated in Fig. 7 (Tabak et al., 2000; van Vreeswijk and Hansel, 2001; Giugliano et al., 2004; Gigante et al., 2007).
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Note that the exact dependence of adaptation on frequency (here given by Eq. (6) and represented as the blue curve in Fig. 7 B) may affect the type of bifurcation that occurs between a silent and an oscillatory network as a source of excitation is increased (Nesse et al., 2008), but it does not change the fundamental mechanism of the oscillations. So the excitatory network can oscillate whether adaptation is input- or output-driven (Nesse et al., 2008). Synaptic adaptation (often called short-term synaptic depression) also produces oscillations in excitatory networks, through the same mechanism as the intrinsic adaptation mechanisms described here (Tabak et al., 2000; Tsodyks et al., 2000; Wiedman and Luthi, 2003). Note, however, that the two types of adaptation (intrinsic or synaptic) can be distinguished in models (even in firing rate models) as well as in experiments based on the models’ predictions (Tabak et al., 2000, 2010; Wiedman and Luthi, 2003). Thus, it is important to pay careful attention to what adaptation model is most appropriate in a particular modeling context. In addition to its rhythmogenic properties in excitatory networks, activity-dependent adaptation facilitates network-based oscillations due to the interaction of excitatory and inhibitory populations (Augustin et al., 2013). The rhythmogenic mechanism based on excitatory connectivity and slow activity-dependent adaptation may underlie various types of network activity such as episodic activity in developing networks (Butts et al., 1999; Tabak et al., 2000), respiratory rhythms (Kosmidis et al., 2004), or slow oscillations between up and down states in cortex (Compte et al., 2003).
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distance requires power law spike rate adaptation. Proc. Natl. Acad. Sci. USA 110: 13624– 13629. Compte A, Sanchez-Vives M, McCormick D, Wang XJ (2003) Cellular and network mechanisms of slow oscillatory activity (