Research paperContributions of adaptation currents to dynamic spike threshold on slow timescales: Biophysical insights from conductance-based models
Introduction
Neurons in the central nervous system (CNS) have powerful capability to process the receiving information underlying perception or motor control. They are able to accurately encode and transmit various spatiotemporal signals by initiating specific sequences of action potentials (APs). One common strategy used by CNS neurons to shape AP trains is the spike-frequency adaptation (SFA), which refers to the gradual reduction of an initially high firing rate to a lower steady-state level during sustained stimulus. Earlier studies have shown that the SFA induced by various adaptation currents plays considerable roles in neural coding, such as signal extraction [1], forward masking [2], [3], response selectivity [4], [5], [6], high-pass filtering [1], [3], [7], [8], noise shaping and precise temporal coding [9], which endows the neuron with a flexible function under varying conditions. Using neuron models with adaptation currents to investigate how the ionic mechanisms underlying SFA participate in AP initiation is therefore an essential step towards a mechanistic understanding of how SFA contributes to the coding strategies of a neuron.
SFA is typically attributed to the slowly hyperpolarizing currents, which are directly or indirectly activated during the APs. Two common types of them include M-type current (IM) [10], [11] and AHP-type current (IAHP) [3], [12], [13]. IM is a voltage-gated K+ current mediated by KCNQ channels. It activates at the potentials below firing threshold [9], [11], [14], and does not rely on AP generation. IAHP is a Ca2+-gated K+ current, which is activated primarily by the increase in intracellular concentration of Ca2+ due to the influx of Ca2+ through voltage-activated Ca2+ channels [3], [9], [12], [13]. The relevant Ca2+ channels are activated at high voltages above firing threshold, which makes the activation of IAHP dependent on the generation of APs. Another adaptation current is the Na+-activated K+ current, i.e., IKNa [15]. It is instantaneously gated by the intracellular concentration of Na+, which can also result in SFA on a much longer timescale. Generally, these inhibitory currents all include a form of slow negative feedback to the excitability of the cell. Their activation usually occurs at a slower timescale than the dynamics of spike generation, which builds up from one AP to the next and then gradually reduces firing rate to constant stimulus.
There is a large variety of single-compartment models used in computational neuroscience to simulate the APs [16], [17], which range from integrate-and-fire (IF) models to various conductance-based models. The former have simplest model forms for spiking activity. In IF model, the AP is emitted when membrane potential exceeds a preset threshold. The common conductance-based models are Hodgkin–Huxley like models, which have more biophysical details than IF neurons. The initiation of APs in these models results from the nonlinear interactions of inward and outward currents at the voltages below firing threshold, i.e., subthreshold. By incorporating various adaptation currents in these single-compartment models, the adaptive behaviors of neurons observed in experiments have been phenomenologically reproduced. Simultaneously, the effects of SFA on the information transmission of neurons have been extensively studied.
For IF model neurons, an alternative way to generate SFA is incorporating a dynamic spike threshold that is incremented by each AP in the model. Earlier theoretical studies [8], [18], [19], [20], [21], [22], [23], [24], [25] have shown that a dynamic threshold is sufficient to lead IF neurons to adapt their response frequency. With such models, the physiologically observed SFA, such as gradually reduced firing rate and negative correlations of interspike intervals (ISIs), has been successfully reproduced. In particular, Liu and Wang [19] compared two methods for reproducing SFA in IF neurons, i.e., adaptation currents and dynamic threshold. They found that there are subtle differences between them when regarding the dependence of either ISI variability or adaptation time constant on input current. A later study by Lindner and Longtin [25] further supports the similarity between two methods. However, Benda et al. [8] recently reported that the adaptation currents have qualitatively different effects on the transfer function of neurons than that of dynamic threshold. These modeling studies that focus on the input-output properties of a neuron lead to the controversy whether the adaptation mechanism should be modeled as a dynamic threshold or as an adaptation current. Is the dynamic spike threshold a possible mechanism for SFA, or just a secondary effect of the activation of adaptation currents?
To address above questions, it requires knowledge of the relationship between adaptation currents and spike threshold. It is known that CNS neurons initiate APs when their membrane voltage exceeds a threshold level [26], [27], i.e., spike threshold. After that, the depolarizing Na+ becomes self-sustaining and drives membrane potential to produce the fast upstroke of the spike. Experiments have shown that the AP threshold is not constant but dynamic [26], [27], [28], [29], [30], which varies with both firing history and synaptic input. In particular, it is inversely correlated with the rate of membrane depolarization, i.e., dV/dt, preceding AP initiation [26], [27], [28], [29], [30], [31], [32]. To determine the relationship between adaptation currents and threshold dynamics, we investigate how the activation of adaptation currents impacts the dependence of threshold voltage on the preceding dV/dt in present study.
Here we use conductance-based models with adaptation currents to simulate SFA. The threshold voltage for AP initiation during the course of SFA is determined by the first derivative method [27], [28], [33] that is widely used in experiments. Instead of characterizing input-output properties, we focus on how adaptation currents interact with depolarizing Na+ at the subthreshold potentials to contribute to the dynamic threshold. These investigations provide a biophysical interpretation of how the intrinsically generated SFA regulates the threshold voltage for AP initiation on slow timescales.
Section snippets
Prescott model
The Prescott model is a single-compartment conductance-based model, which is proposed by introducing adaptation currents to the Morris-Lecar like model [9], [34]. Its state is determined by three dynamical variables, which are membrane potential V, K+ gating variable w, and adaptation variable z. The system is described as follows [9], [34] where Cm is membrane capacitance, Iext is external stimulus, and ϕw is the scale parameter of
Spike threshold in Prescott model without adaptation currents
We start with the Prescott model that is not introduced adaptation currents, i.e., in Eq. (1). In this case, the neuron model only include a depolarizing Na+ and a hyperpolarizing K+ for initiating APs, which does not exhibit SFA to constant stimulus. Here both the firing rate and corresponding rate of membrane depolarization preceding each spike do not decrease as time extends [Fig. 2(a)]. Since there are no inhibitions of slow adaptation current at the subthreshold potentials,
Discussion
Our simulations show that the activations of adaptation currents make the conductance-based model neurons generate a dynamic spike threshold as they reduce firing rate. The adaptation currents in our study include IM, IAHP, and IKNa, which are all slow hyperpolarizing currents. Their activation occurs at a much slower timescale than the fast dynamics of spike generation. It allows an outward current prior to AP initiation to antagonize inward Na+, which then becomes self-sustaining at a more
Conclusion
SFA has been widely studied with regard to its effects on the information processing of neurons. Different from the existing studies, our work has explained a biophysical link between slow adaptation currents and dynamic firing threshold. Such mechanistic investigations elaborate how the intrinsic ionic mechanisms responsible for SFA interact with inward Na+ on a slow timescale to result in threshold dynamics. It highlights the predictive power of membrane biophysics in determining how a neuron
Acknowledgments
This work was supported by grants from the National Natural Science Foundation of China (Nos. 61401312, 61471265 and 61601320), the China Postdoctoral Science Foundation (No. 2015M580202), and the Specialized Research Fund for the Doctoral Program of Higher Education (No. 20130032110065).
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