A multivariate population density model of the dLGN/PGN relay

An Efficient Population Density Method for Modeling Neural Networks with Synaptic Dynamics Manifesting Finite Relaxation Time and Short-Term Plasticity

When incorporating more realistic synaptic dynamics, the computational efficiency of population density methods (PDMs) declines sharply due to the increase in the dimension of master equations. To avoid such a decline, we develop an efficient PDM, termed colored-synapse PDM (csPDM), in which the dimension of the master equations does not depend on the number of synapse-associated state variables in the underlying network model. Our goal is to allow the PDM to incorporate realistic synaptic dynamics that possesses not only finite relaxation time but also short-term plasticity (STP). The model equations of csPDM are derived based on the diffusion approximation on synaptic dynamics and probability density function methods for Langevin equations with colored noise. Numerical examples, given by simulations of the population dynamics of uncoupled exponential integrate-and-fire (EIF) neurons, show good agreement between the results of csPDM and Monte Carlo simulations (MCSs). Compared to the original full-dimensional PDM (fdPDM), the csPDM reveals more excellent computational efficiency because of the lower dimension of the master equations. In addition, it permits network dynamics to possess the short-term plastic characteristics inherited from plastic synapses. The novel csPDM has potential applicability to any spiking neuron models because of no assumptions on neuronal dynamics, and, more importantly, this is the first report of PDM to successfully encompass short-term facilitation/depression properties.

Population Density and Moment-based Approaches to Modeling Domain Calcium-mediated Inactivation of L-type Calcium Channels

We present a population density and moment-based description of the stochastic dynamics of domain [Formula: see text]-mediated inactivation of L-type [Formula: see text] channels. Our approach accounts for the effect of heterogeneity of local [Formula: see text] signals on whole cell [Formula: see text] currents; however, in contrast with prior work, e.g., Sherman et al. (Biophys J 58(4):985-995, 1990), we do not assume that [Formula: see text] domain formation and collapse are fast compared to channel gating. We demonstrate the population density and moment-based modeling approaches using a 12-state Markov chain model of an L-type [Formula: see text] channel introduced by Greenstein and Winslow (Biophys J 83(6):2918-2945, 2002). Simulated whole cell voltage clamp responses yield an inactivation function for the whole cell [Formula: see text] current that agrees with the traditional approach when domain dynamics are fast. We analyze the voltage-dependence of [Formula: see text] inactivation that may occur via slow heterogeneous domain [[Formula: see text]]. Next, we find that when channel permeability is held constant, [Formula: see text]-mediated inactivation of L-type channels increases as the domain time constant increases, because a slow domain collapse rate leads to increased mean domain [[Formula: see text]] near open channels; conversely, when the maximum domain [[Formula: see text]] is held constant, inactivation decreases as the domain time constant increases. Comparison of simulation results using population densities and moment equations confirms the computational efficiency of the moment-based approach, and enables the validation of two distinct methods of truncating and closing the open system of moment equations. In general, a slow domain time constant requires higher order moment truncation for agreement between moment-based and population density simulations.

A principled dimension-reduction method for the population density approach to modeling networks of neurons with synaptic dynamics

The population density approach to neural network modeling has been utilized in a variety of contexts. The idea is to group many similar noisy neurons into populations and track the probability density function for each population that encompasses the proportion of neurons with a particular state rather than simulating individual neurons (i.e., Monte Carlo). It is commonly used for both analytic insight and as a time-saving computational tool. The main shortcoming of this method is that when realistic attributes are incorporated in the underlying neuron model, the dimension of the probability density function increases, leading to intractable equations or, at best, computationally intensive simulations. Thus, developing principled dimension-reduction methods is essential for the robustness of these powerful methods. As a more pragmatic tool, it would be of great value for the larger theoretical neuroscience community. For exposition of this method, we consider a single uncoupled population of leaky integrate-and-fire neurons receiving external excitatory synaptic input only. We present a dimension-reduction method that reduces a two-dimensional partial differential-integral equation to a computationally efficient one-dimensional system and gives qualitatively accurate results in both the steady-state and nonequilibrium regimes. The method, termed modified mean-field method, is based entirely on the governing equations and not on any auxiliary variables or parameters, and it does not require fine-tuning. The principles of the modified mean-field method have potential applicability to more realistic (i.e., higher-dimensional) neural networks.

Calcium window currents, periodic forcing, and chaos: understanding single neuron response with a discontinuous one-dimensional map

Thalamocortical (TC) neurones are known to express the low-voltage activated, inactivating Ca2+ current I(T). The triggering of this current underlies the generation of low threshold Ca2+ potentials that may evoke single or bursts of action potentials. Moreover, this current can contribute to an intrinsic slow (<1 Hz) oscillation whose rhythm is partly determined by the steady state component of I(T) and its interaction with a leak current. This steady state, or window current as it is so often called, has received relatively little theoretical attention despite its importance in determining the electroresponsiveness and input-output relationship of TC neurones. In this paper, we introduce an integrate-and-fire spiking neuron model that includes a biophysically realistic model of I(T). We briefly review the subthreshold bifurcation diagram of this model with constant current injection before moving on to consider its response to periodic forcing. Direct numerical simulations show that as well as the expected mode-locked responses there are regions of parameter space that support chaotic behavior. To reveal the mechanism by which the window current generates a chaotic response to periodic forcing we consider a piecewise linear caricature of the dynamics for the gating variables in the model of I(T). This model can be analyzed in closed form and is shown to support an unstable set of periodic orbits. Trajectories are repelled from these organizing centers until they reach the threshold for firing. By determining the condition for a grazing bifurcation (at the border between a spiking and nonspiking event) we show how knowledge of the unstable periodic orbits (existence and stability) can be combined with the grazing condition to determine an effective one-dimensional map that captures the essentials of the chaotic behavior. This map is discontinuous and has strong similarities with the universal limit mapping in grazing bifurcations derived in the context of impacting mechanical systems [A. B. Nordmark, Phys. Rev. E 55, 266 (1997)].

Models of cardiac excitation-contraction coupling in ventricular myocytes

Mathematical and computational modeling of cardiac excitation-contraction coupling has produced considerable insights into how the heart muscle contracts. With the increase in biophysical and physiological data available, the modeling has become more sophisticated with investigations spanning in scale from molecular components to whole cells. These modeling efforts have provided insight into cardiac excitation-contraction coupling that advanced and complemented experimental studies. One goal is to extend these detailed cellular models to model the whole heart. While this has been done with mechanical and electrophysiological models, the complexity and fast time course of calcium dynamics have made inclusion of detailed calcium dynamics in whole heart models impractical. Novel methods such as the probability density approach and moment closure technique which increase computational efficiency might make this tractable.

Spike train statistics and dynamics with synaptic input from any renewal process: a population density approach

In the probability density function (PDF) approach to neural network modeling, a common simplifying assumption is that the arrival times of elementary postsynaptic events are governed by a Poisson process. This assumption ignores temporal correlations in the input that sometimes have important physiological consequences. We extend PDF methods to models with synaptic event times governed by any modulated renewal process. We focus on the integrate-and-fire neuron with instantaneous synaptic kinetics and a random elementary excitatory postsynaptic potential (EPSP), A. Between presynaptic events, the membrane voltage, v, decays exponentially toward rest, while s, the time since the last synaptic input event, evolves with unit velocity. When a synaptic event arrives, v jumps by A, and s is reset to zero. If v crosses the threshold voltage, an action potential occurs, and v is reset to v(reset). The probability per unit time of a synaptic event at time t, given the elapsed time s since the last event, h(s, t), depends on specifics of the renewal process. We study how regularity of the train of synaptic input events affects output spike rate, PDF and coefficient of variation (CV) of the interspike interval, and the autocorrelation function of the output spike train. In the limit of a deterministic, clocklike train of input events, the PDF of the interspike interval converges to a sum of delta functions, with coefficients determined by the PDF for A. The limiting autocorrelation function of the output spike train is a sum of delta functions whose coefficients fall under a damped oscillatory envelope. When the EPSP CV, sigma A/mu A, is equal to 0.45, a CV for the intersynaptic event interval, sigma T/mu T = 0.35, is functionally equivalent to a deterministic periodic train of synaptic input events (CV = 0) with respect to spike statistics. We discuss the relevance to neural network simulations.

A kinetic theory approach to capturing interneuronal correlation: the feed-forward case

We present an approach for using kinetic theory to capture first and second order statistics of neuronal activity. We coarse grain neuronal networks into populations of neurons and calculate the population average firing rate and output cross-correlation in response to time varying correlated input. We derive coupling equations for the populations based on first and second order statistics of the network connectivity. This coupling scheme is based on the hypothesis that second order statistics of the network connectivity are sufficient to determine second order statistics of neuronal activity. We implement a kinetic theory representation of a simple feed-forward network and demonstrate that the kinetic theory model captures key aspects of the emergence and propagation of correlations in the network, as long as the correlations do not become too strong. By analyzing the correlated activity of feed-forward networks with a variety of connectivity patterns, we provide evidence supporting our hypothesis of the sufficiency of second order connectivity statistics.

Development and validation of a neural population model based on the dynamics of a discontinuous membrane potential neuron model

The major goal of this study was to develop a population density based model derived from statistical mechanics based on the dynamics of a discontinuous membrane potential neuron model. A secondary goal was to validate this model by comparing results from a direct simulation approach on the one hand and our population based approach on the other hand. Comparisons between the two approaches in the case of a synaptically uncoupled and a synaptically coupled neural population produced satisfactory qualitative agreement in terms of firing rate and mean membrane potential. Reasonable quantitative agreement was also obtained for these variables in performed simulations. The results of this work based on the dynamics of a discontinuous membrane potential neuron model provide a basis to simulate phenomenologically large-scale neuronal networks with a reasonably short computing time.

Critical analysis of dimension reduction by a moment closure method in a population density approach to neural network modeling

Computational techniques within the population density function (PDF) framework have provided time-saving alternatives to classical Monte Carlo simulations of neural network activity. Efficiency of the PDF method is lost as the underlying neuron model is made more realistic and the number of state variables increases. In a detailed theoretical and computational study, we elucidate strengths and weaknesses of dimension reduction by a particular moment closure method (Cai, Tao, Shelley, & McLaughlin, 2004; Cai, Tao, Rangan, & McLaughlin, 2006) as applied to integrate-and-fire neurons that receive excitatory synaptic input only. When the unitary postsynaptic conductance event has a single-exponential time course, the evolution equation for the PDF is a partial differential integral equation in two state variables, voltage and excitatory conductance. In the moment closure method, one approximates the conditional kth centered moment of excitatory conductance given voltage by the corresponding unconditioned moment. The result is a system of k coupled partial differential equations with one state variable, voltage, and k coupled ordinary differential equations. Moment closure at k = 2 works well, and at k = 3 works even better, in the regime of high dynamically varying synaptic input rates. Both closures break down at lower synaptic input rates. Phase-plane analysis of the k = 2 problem with typical parameters proves, and reveals why, no steady-state solutions exist below a synaptic input rate that gives a firing rate of 59 s(1) in the full 2D problem. Closure at k = 3 fails for similar reasons. Low firing-rate solutions can be obtained only with parameters for the amplitude or kinetics (or both) of the unitary postsynaptic conductance event that are on the edge of the physiological range. We conclude that this dimension-reduction method gives ill-posed problems for a wide range of physiological parameters, and we suggest future directions.

A probability density approach to modeling local control of calcium-induced calcium release in cardiac myocytes

We present a probability density approach to modeling localized Ca2+ influx via L-type Ca2+ channels and Ca2+-induced Ca2+ release mediated by clusters of ryanodine receptors during excitation-contraction coupling in cardiac myocytes. Coupled advection-reaction equations are derived relating the time-dependent probability density of subsarcolemmal subspace and junctional sarcoplasmic reticulum [Ca2+] conditioned on "Ca2+ release unit" state. When these equations are solved numerically using a high-resolution finite difference scheme and the resulting probability densities are coupled to ordinary differential equations for the bulk myoplasmic and sarcoplasmic reticulum [Ca2+], a realistic but minimal model of cardiac excitation-contraction coupling is produced. Modeling Ca2+ release unit activity using this probability density approach avoids the computationally demanding task of resolving spatial aspects of global Ca2+ signaling, while accurately representing heterogeneous local Ca2+ signals in a population of diadic subspaces and junctional sarcoplasmic reticulum depletion domains. The probability density approach is validated for a physiologically realistic number of Ca2+ release units and benchmarked for computational efficiency by comparison to traditional Monte Carlo simulations. In simulated voltage-clamp protocols, both the probability density and Monte Carlo approaches to modeling local control of excitation-contraction coupling produce high-gain Ca2+ release that is graded with changes in membrane potential, a phenomenon not exhibited by so-called "common pool" models. However, a probability density calculation can be significantly faster than the corresponding Monte Carlo simulation, especially when cellular parameters are such that diadic subspace [Ca2+] is in quasistatic equilibrium with junctional sarcoplasmic reticulum [Ca2+] and, consequently, univariate rather than multivariate probability densities may be employed.