When do correlations increase with firing rates in recurrent networks?

A central question in neuroscience is to understand how noisy firing patterns are used to transmit information. Because neural spiking is noisy, spiking patterns are often quantified via pairwise correlations, or the probability that two cells will spike coincidentally, above and beyond their baseline firing rate. One observation frequently made in experiments, is that correlations can increase systematically with firing rate. Theoretical studies have determined that stimulus-dependent correlations that increase with firing rate can have beneficial effects on information coding; however, we still have an incomplete understanding of what circuit mechanisms do, or do not, produce this correlation-firing rate relationship. Here, we studied the relationship between pairwise correlations and firing rates in recurrently coupled excitatory-inhibitory spiking networks with conductance-based synapses. We found that with stronger excitatory coupling, a positive relationship emerged between pairwise correlations and firing rates. To explain these findings, we used linear response theory to predict the full correlation matrix and to decompose correlations in terms of graph motifs. We then used this decomposition to explain why covariation of correlations with firing rate—a relationship previously explained in feedforward networks driven by correlated input—emerges in some recurrent networks but not in others. Furthermore, when correlations covary with firing rate, this relationship is reflected in low-rank structure in the correlation matrix

One-Dimensional Population Density Approaches to Recurrently Coupled Networks of Neurons with Noise

Mean-field systems have been previously derived for networks of coupled, two-dimensional, integrate-and-fire neurons such as the Izhikevich, adapting exponential (AdEx) and quartic integrate and fire (QIF), among others. Unfortunately, the mean-field systems have a degree of frequency error and the networks analyzed often do not include noise when there is adaptation. Here, we derive a one-dimensional partial differential equation (PDE) approximation for the marginal voltage density under a first order moment closure for coupled networks of integrate-and-fire neurons with white noise inputs. The PDE has substantially less frequency error than the mean-field system, and provides a great deal more information, at the cost of analytical tractability. The convergence properties of the mean-field system in the low noise limit are elucidated. A novel method for the analysis of the stability of the asynchronous tonic firing solution is also presented and implemented. Unlike previous attempts at stability analysis with these network types, information about the marginal densities of the adaptation variables is used. This method can in principle be applied to other systems with nonlinear partial differential equations.Comment: 26 Pages, 6 Figure

Dynamics of Coupled Noisy Neural Oscillators with Heterogeneous Phase Resetting Curves

Pulse-coupled phase oscillators have been utilized in a variety of contexts. Motivated by neuroscience, we study a network of pulse-coupled phase oscillators receiving independent and correlated noise. An additional physiological attribute, heterogeneity, is incorporated in the phase resetting curve (PRC), which is a vital entity for modeling the biophysical dynamics of oscillators. An accurate probability density or mean field description is large dimensional, requiring reduction methods for tractability. We present a reduction method to capture the pairwise synchrony via the probability density of the phase differences, and explore the robustness of the method. We find the reduced methods can capture some of the synchronous dynamics in these networks. The variance of the noisy period (or spike times) in this network is also considered. In particular, we find phase oscillators with predominately positive PRCs (type 1) have larger variance with inhibitory pulse- coupling than PRCs with a larger negative regions (type 2), but with excitatory pulse-coupling the opposite happens – type 1 oscillators have lower variability than type 2. Analysis of this phenomena is provided via an asymptotic approximation with weak noise and weak coupling, where we demonstrate how the individual PRC alters variability with pulse-coupling. We make comparisons of the phase oscillators to full oscillator networks and discuss the utility and shortcomings

Noise-enhanced coding in phasic neuron spike trains

The stochastic nature of neuronal response has lead to conjectures about the impact of input fluctuations on the neural coding. For the most part, low pass membrane integration and spike threshold dynamics have been the primary features assumed in the transfer from synaptic input to output spiking. Phasic neurons are a common, but understudied, neuron class that are characterized by a subthreshold negative feedback that suppresses spike train responses to low frequency signals. Past work has shown that when a low frequency signal is accompanied by moderate intensity broadband noise, phasic neurons spike trains are well locked to the signal. We extend these results with a simple, reduced model of phasic activity that demonstrates that a non-Markovian spike train structure caused by the negative feedback produces a noise-enhanced coding. Further, this enhancement is sensitive to the timescales, as opposed to the intensity, of a driving signal. Reduced hazard function models show that noise-enhanced phasic codes are both novel and separate from classical stochastic resonance reported in non-phasic neurons. The general features of our theory suggest that noise-enhanced codes in excitable systems with subthreshold negative feedback are a particularly rich framework to study

Noise-enhanced coding in phasic neuron spike trains

The stochastic nature of neuronal response has lead to conjectures about the impact of input fluctuations on the neural coding. For the most part, low pass membrane integration and spike threshold dynamics have been the primary features assumed in the transfer from synaptic input to output spiking. Phasic neurons are a common, but understudied, neuron class that are characterized by a subthreshold negative feedback that suppresses spike train responses to low frequency signals. Past work has shown that when a low frequency signal is accompanied by moderate intensity broadband noise, phasic neurons spike trains are well locked to the signal. We extend these results with a simple, reduced model of phasic activity that demonstrates that a non-Markovian spike train structure caused by the negative feedback produces a noise-enhanced coding. Further, this enhancement is sensitive to the timescales, as opposed to the intensity, of a driving signal. Reduced hazard function models show that noise-enhanced phasic codes are both novel and separate from classical stochastic resonance reported in non-phasic neurons. The general features of our theory suggest that noise-enhanced codes in excitable systems with subthreshold negative feedback are a particularly rich framework to study

Analysis of Heterogeneous Cardiac Pacemaker Tissue Models and Traveling Wave Dynamics

The sinoatrial-node (SAN) is a complex heterogeneous tissue that generates a stable rhythm in healthy hearts, yet a general mechanistic explanation for when and how this tissue remains stable is lacking. Although computational and theoretical analyses could elucidate these phenomena, such methods have rarely been used in realistic (large-dimensional) gap-junction coupled heterogeneous pacemaker tissue models. In this study, we adapt a recent model of pacemaker cells (Severi et al. 2012), incorporating biophysical representations of ion channel and intracellular calcium dynamics, to capture physiological features of a heterogeneous population of pacemaker cells, in particular "center" and "peripheral" cells with distinct intrinsic frequencies and action potential morphology. Large-scale simulations of the SAN tissue, represented by a heterogeneous tissue structure of pacemaker cells, exhibit a rich repertoire of behaviors, including complete synchrony, traveling waves of activity originating from periphery to center, and transient traveling waves originating from the center. We use phase reduction methods that do not require fully simulating the large-scale model to capture these observations. Moreover, the phase reduced models accurately predict key properties of the tissue electrical dynamics, including wave frequencies when synchronization occurs, and wave propagation direction in a variety of tissue models. With the reduced phase models, we analyze the relationship between cell distributions and coupling strengths and the resulting transient dynamics. Further, the reduced phase model predicts parameter regimes of irregular electrical dynamics. Thus, we demonstrate that phase reduced oscillator models applied to realistic pacemaker tissue is a useful tool for investigating the spatial-temporal dynamics of cardiac pacemaker activity.Comment: 34 pages, 11 figure

Estimating the Statistics of Operational Loss Through the Analyzation of a Time Series

In the world of finance, appropriately understanding risk is key to success or failure because it is a fundamental driver for institutional behavior. Here we focus on risk as it relates to the operations of financial institutions, namely operational risk. Quantifying operational risk begins with data in the form of a time series of realized losses, which can occur for a number of reasons, can vary over different time intervals, and can pose a challenge that is exacerbated by having to account for both frequency and severity of losses. We introduce a stochastic point process model for the frequency distribution that has two important parameters (average frequency and time scale). The advantages of this model are that the parameters, which we systematically vary to demonstrate accuracy, can be fitted with sufficient data but are also intuitive enough to rely on expert judgment when data is insufficient. Furthermore, we address how to estimate the risk of losses on an arbitrary time scale for a specific frequency model where mathematical techniques can be feasibly applied to analytically calculate the mean, variance, and covariances that are accurate compared to more time-consuming Monte Carlo simulations. Additionally, the auto- and vi cross-correlation functions become mathematically tractable, enabling analytic calculations of cumulative loss statistics over larger time horizons that would otherwise be intractable due to temporal correlations of losses for long time windows. Finally, we demonstrate the strengths and shortcomings of our new approach by using combined data from a consortium of institutions, comparing this data to our model and correlation calculations, and showing that different time horizons can lead to a large range of loss statistics that can significantly affect calculations of capital requirements