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Journal Articles
Publisher: Journals Gateway
Neural Computation (2021) 33 (3): 764–801.
Published: 01 March 2021
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A central theme in computational neuroscience is determining the neural correlates of efficient and accurate coding of sensory signals. Diversity, or heterogeneity, of intrinsic neural attributes is known to exist in many brain areas and is thought to significantly affect neural coding. Recent theoretical and experimental work has argued that in uncoupled networks, coding is most accurate at intermediate levels of heterogeneity. Here we consider this question with data from in vivo recordings of neurons in the electrosensory system of weakly electric fish subject to the same realization of noisy stimuli; we use a generalized linear model (GLM) to assess the accuracy of (Bayesian) decoding of stimulus given a population spiking response. The long recordings enable us to consider many uncoupled networks and a relatively wide range of heterogeneity, as well as many instances of the stimuli, thus enabling us to address this question with statistical power. The GLM decoding is performed on a single long time series of data to mimic realistic conditions rather than using trial-averaged data for better model fits. For a variety of fixed network sizes, we generally find that the optimal levels of heterogeneity are at intermediate values, and this holds in all core components of GLM. These results are robust to several measures of decoding performance, including the absolute value of the error, error weighted by the uncertainty of the estimated stimulus, and the correlation between the actual and estimated stimulus. Although a quadratic fit to decoding performance as a function of heterogeneity is statistically significant, the result is highly variable with low R 2 values. Taken together, intermediate levels of neural heterogeneity are indeed a prominent attribute for efficient coding even within a single time series, but the performance is highly variable.
Journal Articles
Publisher: Journals Gateway
Neural Computation (2013) 25 (10): 2682–2708.
Published: 01 October 2013
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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.
Journal Articles
Publisher: Journals Gateway
Neural Computation (2009) 21 (2): 360–396.
Published: 01 February 2009
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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, σ A /μ A , is equal to 0.45, a CV for the intersynaptic event interval, σ T /μ 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.
Journal Articles
Publisher: Journals Gateway
Neural Computation (2007) 19 (8): 2032–2092.
Published: 01 August 2007
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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 k th 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.