History dependent dynamics in a generic model of ion channels - an analytic study

Frontiers in Computational Neuroscience www.frontiersin.org April 2010 | Volume 4 | Article 3 | 1 COMPUTATIONAL NEUROSCIENCE ORIGINAL RESEARCH ARTICLE published: 08 April 2010 doi: 10.3389/fncom.2010.00003 on transfected oocytes and provides very clear empirical fi ndings. In that experiment a membrane with a population of sodium channels of a single type was clamped at low voltage (−90 mV), then at high voltage (−10 mV), for varying lengths of time – from 10 ms to 5 min. During the high voltage stimulus, the sodium channels entered inac- tivation. Since the fraction of inactivated channels determines the membrane conductivity, by measuring the membrane current, it was possible to observe the dynamics of slow inactivation and recovery in the channel population. After stimulating the membrane with the high voltage clamp for a duration of tstim seconds, the voltage was decreased and clamped back at the low value (−90 mV). At this low voltage level the channels recovered from inactivation. For short stimulations (tstim < 1 s), the recovery was exponential with a single non-history- dependent timescale. After suffi ciently long stimulations (tstim > 1 s) the recovery was distinctly exponential and history dependent, the timescale of recovery monotonically increas- ing with the length of the inactivation period, as seen in Toib et al. (1998). Interestingly, early experiments on visual receptors already displayed similar behavior (Baylor and Hodgkin 1974; in particular see Figures 18A,B). This history-dependence is generally thought to result from the large inactivation state space hinted at by the single channel patch clamp experiments, as suggested fi rst by Toib et al. (1998). Previous modeling approaches, based on this idea, have already been sug- gested in the literature, but fall short in accurately reproducing this behavior. We present a comparative discussion in Section ‘Relation to Previous Work’. One diffi culty in modeling channel behavior is that the nature of the protein conformation dynamics leading to the INTRODUCTION Many 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 (Fairhall et al., 2001; Lundstrom et al., 2008; Wark et al., 2009). At a sub- neuronal level, experiments on sodium (Toib et al., 1998; Melamed-Frank and Marom, 1999; Ellerkmann et al., 2001) and calcium (Uebachs et al., 2006) 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. Finally, patch clamp experiments on single ion channels have hinted at the exist- ence of a large inactivation state space within a single ion channel (Liebovitch and Sullivan, 1987; Millhauser et al., 1988; Marom, 1998 and the references therein). These multi-level experimental fi ndings raise several important questions. How are the behaviors observed at the different levels related (e.g., Lowen et al., 1999)? Specifi cally, is there a connection between the history-dependent timescale of adaptation in the neuron to the history-dependent behavior of ion channels? Does a multitude of inactivation states create the observed channel behavior? What is the functional signifi cance of this his- tory-dependent behavior (e.g., Wark et al., 2009)? Although we do not address all these questions in this paper, we believe that in order to begin addressing them we fi rst need to con- struct a simple working, and mathematically tractable, model of slow inactivation in ion channels. Such a model must reproduce the long term behavior in channel population experiments. Our main focus here is the experiment in Toib et al. (1998), which was performed History-dependent dynamics in a generic model of ion channels – an analytic study Daniel Soudry1,2 and Ron Meir1,2* 1 Department of Electrical Engineering, Technion, Haifa, Israel 2 Laboratory for Network Biology Research, Technion, Haifa, Israel Recent experiments have demonstrated that the timescale of adaptation of single neurons and ion channel populations to stimuli slows down as the length of stimulation increases; in fact, no upper bound on temporal timescales seems to exist in such systems. Furthermore, patch clamp experiments on single ion channels have hinted at the existence of large, mostly unobservable, inactivation state spaces within a single ion channel. This raises the question of the relation between this multitude of inactivation states and the observed behavior. In this work we propose a minimal model for ion channel dynamics which does not assume any specifi c structure of the inactivation state space. The model is simple enough to render an analytical study possible. This leads to a clear and concise explanation of the experimentally obser

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