5 research outputs found
A Lower Bound for the First Passage Time Density of the Suprathreshold Ornstein-Uhlenbeck Process
We prove that the first passage time density for an
Ornstein-Uhlenbeck process obeying to reach
a fixed threshold from a suprathreshold initial condition
has a lower bound of the form for positive constants and for times exceeding some
positive value . We obtain explicit expressions for and in terms
of , , and , and discuss application of the
results to the synchronization of periodically forced stochastic leaky
integrate-and-fire model neurons.Comment: 15 pages, 1 figur
FokkerâPlanck and Fortet Equation-Based Parameter Estimation for a Leaky Integrate-and-Fire Model with Sinusoidal and Stochastic Forcing
Abstract
Analysis of sinusoidal noisy leaky integrate-and-fire models and comparison with experimental data are important to understand the neural code and neural synchronization and rhythms. In this paper, we propose two methods to estimate input parameters using interspike interval data only. One is based on numerical solutions of the FokkerâPlanck equation, and the other is based on an integral equation, which is fulfilled by the interspike interval probability density. This generalizes previous methods tailored to stationary data to the case of time-dependent input. The main contribution is a binning method to circumvent the problems of nonstationarity, and an easy-to-implement initializer for the numerical procedures. The methods are compared on simulated data.
List of Abbreviations
LIF: Leaky integrate-and-fire
ISI: Interspike interval
SDE: Stochastic differential equation
PDE: Partial differential equatio
Graphical modeling of stochastic processes driven by correlated errors
We study a class of graphs that represent local independence structures in
stochastic processes allowing for correlated error processes. Several graphs
may encode the same local independencies and we characterize such equivalence
classes of graphs. In the worst case, the number of conditions in our
characterizations grows superpolynomially as a function of the size of the node
set in the graph. We show that deciding Markov equivalence is coNP-complete
which suggests that our characterizations cannot be improved upon
substantially. We prove a global Markov property in the case of a multivariate
Ornstein-Uhlenbeck process which is driven by correlated Brownian motions.Comment: 43 page
Synchronization in periodically driven and coupled stochastic systems-A discrete state approach
Wir untersuchen das Verhalten von stochastischen bistabilen und erregbaren Systemen auf der Basis einer Modellierung mit diskreten Zuständen. In Ergänzung zum bekannten Markovschen Zwei-Zustandsmodell bistabiler stochastischer Dynamik stellen wir ein nicht Markovsches Drei-Zustandsmodell fĂźr erregbare Systeme vor. Seine relative Einfachheit, verglichen mit stochastischen Modellen erregbarer Dynamik mit kontinuierlichem Phasenraum, ermĂśglicht eine teilweise analytische Auswertung in verschiedenen Zusammenhängen. Zunächst untersuchen wir den gemeinsamen EinfluĂ eines periodischen Treibens und Rauschens. Dieser wird entweder mit Hilfe spektraler GrĂśĂen oder durch Synchronisation des Systems mit dem treibenden Signal charakterisiert. Wir leiten analytische AusdrĂźcke fĂźr die spektrale Leistungsverstärkung und das Signal-zu-Rauschen Verhältnis fĂźr periodisch getriebene Renewal-Prozesse her und wenden diese auf das diskrete Modell fĂźr erregbare Dynamik an. Stochastische Synchronization des Systems mit dem treibenden Signal wird auf der Basis der Diffusionseigenschaften der Ăbergangsereignisse zwischen den diskreten Zuständen untersucht. Wir leiten allgemeine Formeln her, um die mittlere Häufigkeit dieser Ereignisse sowie deren effektiven Diffusionskoeffizienten zu berechnen. Ăber die konkrete Anwendung auf die untersuchten diskreten Modelle hinaus stellen diese Ergebnisse ein neues Werkzeug fĂźr die Untersuchung periodischer Renewal-Prozesse dar. SchlieĂlich betrachten wir noch das Verhalten global gekoppelter bistabiler und erregbarer Systeme. Im Gegensatz zu bistabilen System kĂśnnen erregbare Systeme synchronisiert werden und zeigen kohärente Oszillationen. Alle Untersuchungen des nicht Markovschen Drei-Zustandsmodells werden mit dem prototypischen Modell fĂźr erregbare Dynamik, dem FitzHugh-Nagumo System, verglichen und zeigen eine gute Ăbereinstimmung.We investigate the behavior of stochastic bistable and excitable dynamics based on a discrete state modeling. In addition to the well known Markovian two state model for bistable dynamics we introduce a non Markovian three state model for excitable systems. Its relative simplicity compared to stochastic models of excitable dynamics with continuous phase space allows to obtain analytical results in different contexts. First, we study the joint influence of periodic signals and noise, both based on a characterization in terms of spectral quantities and in terms of synchronization with the periodic driving. We present expressions for the spectral power amplification and signal to noise ratio for renewal processes driven by periodic signals and apply these results to the discrete model for excitable systems. Stochastic synchronization of the system to the driving signal is investigated based on diffusion properties of the transition events between the discrete states. We derive general results for the mean frequency and effective diffusion coefficient which, beyond the application to the discrete models considered in this work, provide a new tool in the study of periodically driven renewal processes. Finally the behavior of globally coupled excitable and bistable units is investigated based on the discrete state description. In contrast to the bistable systems, the excitable system exhibits synchronization and thus coherent oscillations. All investigations of the non Markovian three state model are compared with the prototypical continuous model for excitable dynamics, the FitzHugh-Nagumo system, revealing a good agreement between both models
Sample Path Analysis of Integrate-and-Fire Neurons
Computational neuroscience is concerned with answering two intertwined questions that are based on the assumption that spatio-temporal patterns of spikes form the universal language of the nervous system. First, what function does a specific neural circuitry perform in the elaboration of a behavior? Second, how do neural circuits process behaviorally-relevant information? Non-linear system analysis has proven instrumental in understanding the coding strategies of early neural processing in various sensory modalities. Yet, at higher levels of integration, it fails to help in deciphering the response of assemblies of neurons to complex naturalistic stimuli. If neural activity can be assumed to be primarily driven by the stimulus at early stages of processing, the intrinsic activity of neural circuits interacts with their high-dimensional input to transform it in a stochastic non-linear fashion at the cortical level. As a consequence, any attempt to fully understand the brain through a system analysis approach becomes illusory. However, it is increasingly advocated that neural noise plays a constructive role in neural processing, facilitating information transmission. This prompts to gain insight into the neural code by studying the stochasticity of neuronal activity, which is viewed as biologically relevant. Such an endeavor requires the design of guiding theoretical principles to assess the potential benefits of neural noise. In this context, meeting the requirements of biological relevance and computational tractability, while providing a stochastic description of neural activity, prescribes the adoption of the integrate-and-fire model. In this thesis, founding ourselves on the path-wise description of neuronal activity, we propose to further the stochastic analysis of the integrate-and fire model through a combination of numerical and theoretical techniques. To begin, we expand upon the path-wise construction of linear diffusions, which offers a natural setting to describe leaky integrate-and-fire neurons, as inhomogeneous Markov chains. Based on the theoretical analysis of the first-passage problem, we then explore the interplay between the internal neuronal noise and the statistics of injected perturbations at the single unit level, and examine its implications on the neural coding. At the population level, we also develop an exact event-driven implementation of a Markov network of perfect integrate-and-fire neurons with both time delayed instantaneous interactions and arbitrary topology. We hope our approach will provide new paradigms to understand how sensory inputs perturb neural intrinsic activity and accomplish the goal of developing a new technique for identifying relevant patterns of population activity. From a perturbative perspective, our study shows how injecting frozen noise in different flavors can help characterize internal neuronal noise, which is presumably functionally relevant to information processing. From a simulation perspective, our event-driven framework is amenable to scrutinize the stochastic behavior of simple recurrent motifs as well as temporal dynamics of large scale networks under spike-timing-dependent plasticity