Analytical study of history dependent timescales in a generic model of ion channels
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Technion , Electrical Engineering, Israel
Recent experiments have demonstrated that the timescale of adaptation of a single neuron in response to periodic stimuli slows down as the period of stimulation increases. At a sub-neuronal level, experiments on sodium and calcium ion channel populations have shown that the timescale of the recovery from inactivation following a long duration of membrane depolarization increased with the length of the depolarization period. We refer to this type of behavior as history-dependence. The origin of this history dependence is generally thought to result from the large inactivation state space hinted at by single channel patch clamp experiments. Previous modeling approaches, based on this idea, have already been suggested in the literature, but fall short in accurately reproducing this behavior. We model the slow inactivation of a channel as a continuous-time semi-Markov process consisting only of two states, the Markovian "available" state, and the non-Markovian "inactivated" state. The residence time probability density function (RTPDF) of the Markovian available state is exponential, while the inactivated state is non-Markovian, with a power-law RTPDF. Both RTPDFs are voltage dependent. These channel RTPDFs are measurable quantities. We reproduce for the first time, to our knowledge, the main experimental finding observed in the channel populations experiments, namely an exponential recovery process with a history-dependent timescale. Using these results we narrow down the options for the model parameters at different voltages, and explicitly address the issue of long memory phenomena. The model introduced here also provides many predictions. Qualitatively, we predict that spiking stimuli change the timescale of channel recovery from inactivation only negligibly, and that the rate of this recovery must be voltage dependent. Quantitatively, we derive an exact dynamic equation that fully defines an input-output relation between the membrane voltage and the channel availability, and solve it exactly in many important cases. Additionally, we develop expressions that describe all joint moments in the single channel and population. The potential contribution of this model goes beyond the specific system addressed in this work. Current models of channels and receptors tend to suffer from an embarrassment of riches. In order to explain behaviors over an ever expanding range of timescales, these complex models often include multiple inactivation states. Since the number of states and their parameters are not directly observable, these models tend to be highly specific and are likely to suffer from over-fitting. Furthermore, such models always have an upper bound on their timescale. In this work, we introduce and thoroughly analyze a type of model that does not suffer from these limitations. Despite its simplicity, it provides a generalization of previous models, is based only on measurable quantities, does not possess an upper bound on its timescale and exhibits considerable analytical tractability. As such, it stands as an appealing alternative to previous approaches. In particular, given the direct impact of channel dynamics on neuronal behavior, we expect that this experimentally well motivated, yet mathematically tractable model will form a sound foundation for realistic neuronal models, spanning multiple time scales and exhibiting history-dependence.
Conference:
Computational and Systems Neuroscience 2010, Salt Lake City, UT, United States, 25 Feb - 2 Mar, 2010.
Presentation Type:
Poster Presentation
Topic:
Poster session I
Citation:
Soudry
D and
Meir
R
(2010). Analytical study of history dependent timescales in a generic model of ion channels.
Front. Neurosci.
Conference Abstract:
Computational and Systems Neuroscience 2010.
doi: 10.3389/conf.fnins.2010.03.00007
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Received:
17 Feb 2010;
Published Online:
17 Feb 2010.
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Correspondence:
Ron Meir, Technion, Electrical Engineering, Haifa, Israel, rmeir@ee.technion.ac.il