Correlations in Ion Channel Expression Emerge from PNAS PLUS Homeostatic Tuning Rules
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Correlations in ion channel expression emerge from PNAS PLUS homeostatic tuning rules Timothy O’Leary1, Alex H. Williams, Jonathan S. Caplan, and Eve Marder1 Volen Center for Complex Systems, Department of Biology, Brandeis University, Waltham, MA 02454 Contributed by Eve Marder, May 29, 2013 (sent for review April 24, 2013) Experimental observations reveal that the expression levels of dif- a homeostatic tuning rule) were made from random sets of con- ferent ion channels vary across neurons of a defined type, even ductance parameters and then searched for those that produce a when these neurons exhibit stereotyped electrical properties. How- specific behavior, they did not show correlations in conductance ever, there are robust correlations between different ion channel expression that resemble the experimental findings (37). This find- expression levels, although the mechanisms that determine these ing raised the question of how the correlations seen in the experi- correlations are unknown. Using generic model neurons, we show mental data are established and whether they are somehow that correlated conductance expression can emerge from simple homeostatic control mechanisms that couple expression rates of genetically hardwired. For example, correlations in ion channel individual conductances to cellular readouts of activity. The correla- expression may simply result from explicit coregulation, such as tions depend on the relative rates of expression of different con- control of gene expression by a common transcription factor or ductances. Thus, variability is consistent with homeostatic regulation silencing of a subset of genes in a certain population of cells. and the structure of this variability reveals quantitative relations be- Another possibility is that correlations emerge from some in- tween regulation dynamics of different conductances. Furthermore, teraction between activity-dependent regulatory processes that we show that homeostatic regulation is remarkably insensitive to the control the expression of different ion channel types. details that couple the regulation of a given conductance to overall We address this question in this paper using theory and compu- neuronal activity because of degeneracy in the function of multiple tational models. We show that correlations in ion channel expres- conductances and can be robust to “antihomeostatic” regulation of sion emerge as a consequence of homeostatic control mechanisms NEUROSCIENCE a subset of conductances expressed in a cell. that couple the expression rates of individual conductances to a cell- intrinsic readout of activity. Importantly, the shape of the correla- neuronal excitability | robustness | computational models | control theory tion pattern is determined by the relative rates of expression of different conductances. Furthermore, we show how degeneracy he electrophysiological signature of every neuron is determined Tby the number and kind of voltage-dependent conductances implies that regulatory control mechanisms do not need to be as in its membrane. Most neurons express many voltage-dependent precisely tuned as previously anticipated. For example, subsets of conductances, some of which may have overlapping or degenerate conductances can be regulated antihomeostatically without in- physiological functions (1–6). Furthermore, neurons in the brains terfering with convergence to a target activity level, meaning that of long-lived animals must maintain reliable function over the inward conductances can up-regulate in response to elevated animal’s lifetime while all of their ion channels and receptors are activity and vice versa for outward conductances. Thus, there is replaced in the membrane over hours, days, or weeks. Conse- considerable flexibility in how different conductances can be quently, ongoing turnover of ion channels of various types must regulated while maintaining a “set point” in activity. This flexi- occur without compromising the essential excitability properties bility is compatible with distinct correlation patterns seen in the – of the neuron (5, 7 10). conductance expression of different neuron types. Both theoretical and experimental studies suggest that main- taining stable intrinsic excitability is accomplished via homeostatic, 2+ negative feedback processes that use intracellular Ca concen- Significance trations as a sensor of activity and then alter the synthesis, in- sertion, and degradation of membrane conductances to achieve a target activity level (11–27). Among the modeling studies are A deep puzzle in neuroscience is how neurons maintain their electrical properties despite continuous ion channel turnover several different homeostatic tuning rules that differ in how sensor and activity perturbations. Previous work proposed that ac- readout is coupled to the changes in conductance necessary to tivity-dependent homeostatic rules ensure robust development achieve a target activity (11, 13, 14, 28). Regardless, these models of excitability by regulating channel density, although it is not can self-assemble from randomized initial conditions, and they will understood how these rules shape the distribution of ion change their conductance densities in response to perturbation or channel types nor how finely tuned these rules must be. We synaptic drive. In one of these homeostatic self-tuning models (14), show that generic homeostatic regulation rules impose corre- similar activity patterns can be associated with different sets of lations in the steady-state distribution of ion channels, as has conductance densities. been recently observed experimentally. Specific correlations Thus, it is perhaps not surprising that experimental studies also fi depend on relative expression rates, and the regulation rules nd a considerable range in the conductance densities of voltage- themselves are far more robust than previously thought. dependent channels and in the mRNA expression of their ion – channel genes (29 36). The experimental studies also showed clear Author contributions: T.O. and E.M. designed research; T.O., A.H.W., and J.S.C. performed correlations in these expression patterns (30, 32–35); for example, research; T.O., A.H.W., and J.S.C. analyzed data; and T.O. and E.M. wrote the paper. strong linear correlations are found between mRNA copy number The authors declare no conflict of interest. for shal/A-type potassium channels and IH/hyperpolarization/ Freely available online through the PNAS open access option. fi cyclic nucleotide activated channels in identi ed crustacean 1To whom correspondence may be addressed. E-mail: [email protected] or marder@ motor neurons. It is therefore possible that these correlations are brandeis.edu. crucial for the electrophysiological behavior of the neuron in This article contains supporting information online at www.pnas.org/lookup/suppl/doi:10. question. However, when large numbers of model neurons (without 1073/pnas.1309966110/-/DCSupplemental. www.pnas.org/cgi/doi/10.1073/pnas.1309966110 PNAS Early Edition | 1of10 Downloaded by guest on September 26, 2021 Results steeper rates of change in this case. In both of these versions of the 2+ There are several existing homeostatic neuron models that use model, [Ca ] equilibrates at its target value. Interestingly, 2+ intracellular Ca concentrations to regulate their conductances when we changed the sign of g2 (thus making its direction of (11, 13, 14, 18, 28, 38). These models are capable of producing and regulation antihomeostatic), the model also converges to target 2+ maintaining complex activity patterns such as rhythmic bursting [Ca ] value. In fact, homeostatic models with multiple con- that rely on the interactions between many voltage-dependent ductances can tolerate such antihomeostatic regulation in conductances and Ca2+ dynamics. Analysis of these models is of- a subset of conductances provided broad constraints on the τ ten mathematically intractable, and it is also difficult to develop an regulation rates ( i) are respected. fl intuitive understanding of how the distribution of conductances How do the regulation rates in uence the resulting steady-state is shaped over time. Therefore, in this study we start with a toy distribution of conductances? Fig. 1D shows three views of a 3D model with three non–voltage-dependent conductances and simple plot showing the conductances as they are distributed initially (or- Ca2+ dynamics. We then progress to a more complicated spiking ange points) and at steady state. Each version of the model (with model with three regulated voltage-dependent conductances and different sets of regulation rates) converges to a distinct region of finish with an analysis of an existing model that has seven voltage- conductance space, but these regions sit on a common plane (pink dependent conductances and three distinct [Ca2+] sensors. In all rectangle). This plane is simply the solution set of all conductances three cases, we examine how the correlations in the steady-state that produce target activity in the model. Thus, the regulation rates conductance distributions are shaped by the parameters that (as well as the initial values of the conductances) determine the govern regulation. We find that the intuition developed in the direction in which the model evolves in conductance space, whereas simplest model carries over to more complex cases. the point of intersection of each trajectory with the solution plane dictates the steady-state conductance values. Correlations Arise in Simple Model of Homeostatic The correlation between