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Layer M, Senk J, Essink S, van Meegen A, Bos H, Helias M. NNMT: Mean-Field Based Analysis Tools for Neuronal Network Models. Front Neuroinform 2022; 16:835657. [PMID: 35712677 PMCID: PMC9196133 DOI: 10.3389/fninf.2022.835657] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 12/14/2021] [Accepted: 03/17/2022] [Indexed: 11/13/2022] Open
Abstract
Mean-field theory of neuronal networks has led to numerous advances in our analytical and intuitive understanding of their dynamics during the past decades. In order to make mean-field based analysis tools more accessible, we implemented an extensible, easy-to-use open-source Python toolbox that collects a variety of mean-field methods for the leaky integrate-and-fire neuron model. The Neuronal Network Mean-field Toolbox (NNMT) in its current state allows for estimating properties of large neuronal networks, such as firing rates, power spectra, and dynamical stability in mean-field and linear response approximation, without running simulations. In this article, we describe how the toolbox is implemented, show how it is used to reproduce results of previous studies, and discuss different use-cases, such as parameter space explorations, or mapping different network models. Although the initial version of the toolbox focuses on methods for leaky integrate-and-fire neurons, its structure is designed to be open and extensible. It aims to provide a platform for collecting analytical methods for neuronal network model analysis, such that the neuroscientific community can take maximal advantage of them.
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Affiliation(s)
- Moritz Layer
- Institute of Neuroscience and Medicine (INM-6) and Institute for Advanced Simulation (IAS-6) and JARA-Institute Brain Structure-Function Relationships (INM-10), Jülich Research Centre, Jülich, Germany
- RWTH Aachen University, Aachen, Germany
| | - Johanna Senk
- Institute of Neuroscience and Medicine (INM-6) and Institute for Advanced Simulation (IAS-6) and JARA-Institute Brain Structure-Function Relationships (INM-10), Jülich Research Centre, Jülich, Germany
| | - Simon Essink
- Institute of Neuroscience and Medicine (INM-6) and Institute for Advanced Simulation (IAS-6) and JARA-Institute Brain Structure-Function Relationships (INM-10), Jülich Research Centre, Jülich, Germany
- RWTH Aachen University, Aachen, Germany
| | - Alexander van Meegen
- Institute of Neuroscience and Medicine (INM-6) and Institute for Advanced Simulation (IAS-6) and JARA-Institute Brain Structure-Function Relationships (INM-10), Jülich Research Centre, Jülich, Germany
- Institute of Zoology, Faculty of Mathematics and Natural Sciences, University of Cologne, Cologne, Germany
| | - Hannah Bos
- Institute of Neuroscience and Medicine (INM-6) and Institute for Advanced Simulation (IAS-6) and JARA-Institute Brain Structure-Function Relationships (INM-10), Jülich Research Centre, Jülich, Germany
| | - Moritz Helias
- Institute of Neuroscience and Medicine (INM-6) and Institute for Advanced Simulation (IAS-6) and JARA-Institute Brain Structure-Function Relationships (INM-10), Jülich Research Centre, Jülich, Germany
- Department of Physics, Faculty 1, RWTH Aachen University, Aachen, Germany
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2
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Bernardi D, Lindner B. Receiver operating characteristic curves for a simple stochastic process that carries a static signal. Phys Rev E 2020; 101:062132. [PMID: 32688497 DOI: 10.1103/physreve.101.062132] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.3] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 04/17/2020] [Accepted: 05/14/2020] [Indexed: 11/07/2022]
Abstract
The detection of a weak signal in the presence of noise is an important problem that is often studied in terms of the receiver operating characteristic (ROC) curve, in which the probability of correct detection is plotted against the probability for a false positive. This kind of analysis is typically applied to the situation in which signal and noise are stochastic variables; the detection problem emerges, however, also often in a context in which both signal and noise are stochastic processes and the (correct or false) detection is said to take place when the process crosses a threshold in a given time window. Here we consider the problem for a combination of a static signal which has to be detected against a dynamic noise process, the well-known Ornstein-Uhlenbeck process. We give exact (but difficult to evaluate) quadrature expressions for the detection rates for false positives and correct detections, investigate systematically a simple sampling approximation suggested earlier, compare to an approximation by Stratonovich for the limit of high threshold, and briefly explore the case of multiplicative signal; all theoretical results are compared to extensive numerical simulations of the corresponding Langevin equation. Our results demonstrate that the sampling approximation provides a reasonable description of the ROC curve for this system, and it clarifies limit cases for the ROC curve.
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Affiliation(s)
- Davide Bernardi
- Bernstein Center for Computational Neuroscience Berlin, 10115 Berlin, Germany and Physics Department of Humboldt University Berlin, 12489 Berlin, Germany
| | - Benjamin Lindner
- Bernstein Center for Computational Neuroscience Berlin, 10115 Berlin, Germany and Physics Department of Humboldt University Berlin, 12489 Berlin, Germany
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3
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Stapmanns J, Kühn T, Dahmen D, Luu T, Honerkamp C, Helias M. Self-consistent formulations for stochastic nonlinear neuronal dynamics. Phys Rev E 2020; 101:042124. [PMID: 32422832 DOI: 10.1103/physreve.101.042124] [Citation(s) in RCA: 5] [Impact Index Per Article: 1.3] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 01/16/2019] [Accepted: 12/18/2019] [Indexed: 01/28/2023]
Abstract
Neural dynamics is often investigated with tools from bifurcation theory. However, many neuron models are stochastic, mimicking fluctuations in the input from unknown parts of the brain or the spiking nature of signals. Noise changes the dynamics with respect to the deterministic model; in particular classical bifurcation theory cannot be applied. We formulate the stochastic neuron dynamics in the Martin-Siggia-Rose de Dominicis-Janssen (MSRDJ) formalism and present the fluctuation expansion of the effective action and the functional renormalization group (fRG) as two systematic ways to incorporate corrections to the mean dynamics and time-dependent statistics due to fluctuations in the presence of nonlinear neuronal gain. To formulate self-consistency equations, we derive a fundamental link between the effective action in the Onsager-Machlup (OM) formalism, which allows the study of phase transitions, and the MSRDJ effective action, which is computationally advantageous. These results in particular allow the derivation of an OM effective action for systems with non-Gaussian noise. This approach naturally leads to effective deterministic equations for the first moment of the stochastic system; they explain how nonlinearities and noise cooperate to produce memory effects. Moreover, the MSRDJ formulation yields an effective linear system that has identical power spectra and linear response. Starting from the better known loopwise approximation, we then discuss the use of the fRG as a method to obtain self-consistency beyond the mean. We present a new efficient truncation scheme for the hierarchy of flow equations for the vertex functions by adapting the Blaizot, Méndez, and Wschebor approximation from the derivative expansion to the vertex expansion. The methods are presented by means of the simplest possible example of a stochastic differential equation that has generic features of neuronal dynamics.
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Affiliation(s)
- Jonas Stapmanns
- Institute of Neuroscience and Medicine (INM-6) and Institute for Advanced Simulation (IAS-6) and JARA BRAIN Institute I, Jülich Research Centre, Jülich, Germany.,Institute for Theoretical Solid State Physics, RWTH Aachen University, 52074 Aachen, Germany
| | - Tobias Kühn
- Institute of Neuroscience and Medicine (INM-6) and Institute for Advanced Simulation (IAS-6) and JARA BRAIN Institute I, Jülich Research Centre, Jülich, Germany.,Institute for Theoretical Solid State Physics, RWTH Aachen University, 52074 Aachen, Germany
| | - David Dahmen
- Institute of Neuroscience and Medicine (INM-6) and Institute for Advanced Simulation (IAS-6) and JARA BRAIN Institute I, Jülich Research Centre, Jülich, Germany
| | - Thomas Luu
- Institut für Kernphysik (IKP-3), Institute for Advanced Simulation (IAS-4) and Jülich Center for Hadron Physics, Jülich Research Centre, Jülich, Germany
| | - Carsten Honerkamp
- Institute for Theoretical Solid State Physics, RWTH Aachen University, 52074 Aachen, Germany.,JARA-FIT, Jülich Aachen Research Alliance-Fundamentals of Future Information Technology, Germany
| | - Moritz Helias
- Institute of Neuroscience and Medicine (INM-6) and Institute for Advanced Simulation (IAS-6) and JARA BRAIN Institute I, Jülich Research Centre, Jülich, Germany.,Institute for Theoretical Solid State Physics, RWTH Aachen University, 52074 Aachen, Germany
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4
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Dubey A, Bandyopadhyay M. DNA breathing dynamics under periodic forcing: Study of several distribution functions of relevant Brownian functionals. Phys Rev E 2019; 100:052107. [PMID: 31869881 DOI: 10.1103/physreve.100.052107] [Citation(s) in RCA: 2] [Impact Index Per Article: 0.4] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 07/25/2019] [Indexed: 06/10/2023]
Abstract
In this paper, we study DNA breathing dynamics in the presence of an external periodic force by proposing and inspecting several probability distribution functions (PDFs) of relevant Brownian functionals which specify the bubble lifetime, reactivity, and average size. We model the bubble dynamics process by an overdamped Langevin equation of broken base pairs on the Poland-Scheraga free energy landscape. Introducing an effective time-independent description for timescales larger than T[over ̃]=2π/ω (where ω is the frequency of external periodic force) and using an elegant backward Fokker-Planck method we derive closed form expressions of several PDFs associated with such stochastic processes. For instance, with an initial bubble size of x_{0}, we derive the following analytical expressions: (i) the PDF P(t_{f}|x_{0}) of the first passage time t_{f} which specifies the lifetime of the DNA breathing process, (ii) the PDF P(A|x_{0}) of the area A until the first passage time, and it provides much valuable information about the average bubble size and reactivity of the process, and (iii) the PDF P(M) associated with the maximum bubble size M of the breathing process before complete denaturation. Our analysis is limited to two limits: (a) large bubble size and (b) small bubble size. We further confirm our analytical predictions by computing the same PDFs with direct numerical simulations of the corresponding Langevin equations. We obtain very good agreement of our theoretical predictions with the numerically simulated results. Finally, several nontrivial scaling behaviors in the asymptotic limits for the above-mentioned PDFs are predicted, which can be verified further from experimental observation. Our main conclusion is that the large bubble dynamics is unaffected by the rapidly oscillating force, but the small bubble dynamics is significantly affected by the same periodic force.
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Affiliation(s)
- Ashutosh Dubey
- School of Basic Sciences, Indian Institute of Technology Bhubaneswar, Bhubaneswar 751007, India
| | - Malay Bandyopadhyay
- School of Basic Sciences, Indian Institute of Technology Bhubaneswar, Bhubaneswar 751007, India
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5
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Braun W, Thul R, Longtin A. Evolution of moments and correlations in nonrenewal escape-time processes. Phys Rev E 2017; 95:052127. [PMID: 28618562 DOI: 10.1103/physreve.95.052127] [Citation(s) in RCA: 5] [Impact Index Per Article: 0.7] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 08/05/2016] [Indexed: 06/07/2023]
Abstract
The theoretical description of nonrenewal stochastic systems is a challenge. Analytical results are often not available or can be obtained only under strong conditions, limiting their applicability. Also, numerical results have mostly been obtained by ad hoc Monte Carlo simulations, which are usually computationally expensive when a high degree of accuracy is needed. To gain quantitative insight into these systems under general conditions, we here introduce a numerical iterated first-passage time approach based on solving the time-dependent Fokker-Planck equation (FPE) to describe the statistics of nonrenewal stochastic systems. We illustrate the approach using spike-triggered neuronal adaptation in the leaky and perfect integrate-and-fire model, respectively. The transition to stationarity of first-passage time moments and their sequential correlations occur on a nontrivial time scale that depends on all system parameters. Surprisingly this is so for both single exponential and scale-free power-law adaptation. The method works beyond the small noise and time-scale separation approximations. It shows excellent agreement with direct Monte Carlo simulations, which allow for the computation of transient and stationary distributions. We compare different methods to compute the evolution of the moments and serial correlation coefficients (SCCs) and discuss the challenge of reliably computing the SCCs, which we find to be very sensitive to numerical inaccuracies for both the leaky and perfect integrate-and-fire models. In conclusion, our methods provide a general picture of nonrenewal dynamics in a wide range of stochastic systems exhibiting short- and long-range correlations.
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Affiliation(s)
- Wilhelm Braun
- Department of Physics and Centre for Neural Dynamics, University of Ottawa, 598 King Edward, Ottawa K1N 6N5, Canada
- University of Ottawa Brain and Mind Research Institute, University of Ottawa, 451 Smyth Road, Ottawa, ON K1H 8M5, Canada
| | - Rüdiger Thul
- Centre for Mathematical Medicine and Biology, School of Mathematical Sciences, University of Nottingham, Nottingham NG7 2RD, United Kingdom
| | - André Longtin
- Department of Physics and Centre for Neural Dynamics, University of Ottawa, 598 King Edward, Ottawa K1N 6N5, Canada
- University of Ottawa Brain and Mind Research Institute, University of Ottawa, 451 Smyth Road, Ottawa, ON K1H 8M5, Canada
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Tamborrino M. Approximation of the first passage time density of a Wiener process to an exponentially decaying boundary by two-piecewise linear threshold. Application to neuronal spiking activity. MATHEMATICAL BIOSCIENCES AND ENGINEERING : MBE 2016; 13:613-629. [PMID: 27106189 DOI: 10.3934/mbe.2016011] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.1] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 06/05/2023]
Abstract
The first passage time density of a diffusion process to a time varying threshold is of primary interest in different fields. Here, we consider a Brownian motion in presence of an exponentially decaying threshold to model the neuronal spiking activity. Since analytical expressions of the first passage time density are not available, we propose to approximate the curved boundary by means of a continuous two-piecewise linear threshold. Explicit expressions for the first passage time density towards the new boundary are provided. First, we introduce different approximating linear thresholds. Then, we describe how to choose the optimal one minimizing the distance to the curved boundary, and hence the error in the corresponding passage time density. Theoretical means, variances and coefficients of variation given by our method are compared with empirical quantities from simulated data. Moreover, a further comparison with firing statistics derived under the assumption of a small amplitude of the time-dependent change in the threshold, is also carried out. Finally, maximum likelihood and moment estimators of the parameters of the model are derived and applied on simulated data.
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7
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Kobayashi R, Kitano K. Impact of slow K(+) currents on spike generation can be described by an adaptive threshold model. J Comput Neurosci 2016; 40:347-62. [PMID: 27085337 PMCID: PMC4860204 DOI: 10.1007/s10827-016-0601-0] [Citation(s) in RCA: 16] [Impact Index Per Article: 2.0] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 02/26/2015] [Revised: 03/06/2016] [Accepted: 04/01/2016] [Indexed: 12/01/2022]
Abstract
A neuron that is stimulated by rectangular current injections initially responds with a high firing rate, followed by a decrease in the firing rate. This phenomenon is called spike-frequency adaptation and is usually mediated by slow K(+) currents, such as the M-type K(+) current (I M ) or the Ca(2+)-activated K(+) current (I AHP ). It is not clear how the detailed biophysical mechanisms regulate spike generation in a cortical neuron. In this study, we investigated the impact of slow K(+) currents on spike generation mechanism by reducing a detailed conductance-based neuron model. We showed that the detailed model can be reduced to a multi-timescale adaptive threshold model, and derived the formulae that describe the relationship between slow K(+) current parameters and reduced model parameters. Our analysis of the reduced model suggests that slow K(+) currents have a differential effect on the noise tolerance in neural coding.
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Affiliation(s)
- Ryota Kobayashi
- Principles of Informatics Research Division, National Institute of Informatics, 2-1-2 Hitotsubashi, Chiyoda-ku, Tokyo, Japan. .,Department of Informatics, SOKENDAI (The Graduate University for Advanced Studies), 2-1-2 Hitotsubashi, Chiyoda-ku, Tokyo, Japan.
| | - Katsunori Kitano
- Department of Human and Computer Intelligence, Ritsumeikan University, 1-1-1 Nojihigashi, Kusatsu, Shiga, 525-8577, Japan
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8
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Braun W, Matthews PC, Thul R. First-passage times in integrate-and-fire neurons with stochastic thresholds. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2015; 91:052701. [PMID: 26066193 DOI: 10.1103/physreve.91.052701] [Citation(s) in RCA: 2] [Impact Index Per Article: 0.2] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Subscribe] [Scholar Register] [Received: 12/10/2014] [Indexed: 06/04/2023]
Abstract
We consider a leaky integrate-and-fire neuron with deterministic subthreshold dynamics and a firing threshold that evolves as an Ornstein-Uhlenbeck process. The formulation of this minimal model is motivated by the experimentally observed widespread variation of neural firing thresholds. We show numerically that the mean first-passage time can depend nonmonotonically on the noise amplitude. For sufficiently large values of the correlation time of the stochastic threshold the mean first-passage time is maximal for nonvanishing noise. We provide an explanation for this effect by analytically transforming the original model into a first-passage-time problem for Brownian motion. This transformation also allows for a perturbative calculation of the first-passage-time histograms. In turn this provides quantitative insights into the mechanisms that lead to the nonmonotonic behavior of the mean first-passage time. The perturbation expansion is in excellent agreement with direct numerical simulations. The approach developed here can be applied to any deterministic subthreshold dynamics and any Gauss-Markov processes for the firing threshold. This opens up the possibility to incorporate biophysically detailed components into the subthreshold dynamics, rendering our approach a powerful framework that sits between traditional integrate-and-fire models and complex mechanistic descriptions of neural dynamics.
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Affiliation(s)
- Wilhelm Braun
- School of Mathematical Sciences and Centre for Mathematical Medicine and Biology, University of Nottingham, University Park, Nottingham, NG7 2RD, United Kingdom
| | - Paul C Matthews
- School of Mathematical Sciences and Centre for Mathematical Medicine and Biology, University of Nottingham, University Park, Nottingham, NG7 2RD, United Kingdom
| | - Rüdiger Thul
- School of Mathematical Sciences and Centre for Mathematical Medicine and Biology, University of Nottingham, University Park, Nottingham, NG7 2RD, United Kingdom
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9
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Kim H, Shinomoto S. Estimating nonstationary inputs from a single spike train based on a neuron model with adaptation. MATHEMATICAL BIOSCIENCES AND ENGINEERING : MBE 2014; 11:49-62. [PMID: 24245682 DOI: 10.3934/mbe.2014.11.49] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.1] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 06/02/2023]
Abstract
Because every spike of a neuron is determined by input signals, a train of spikes may contain information about the dynamics of unobserved neurons. A state-space method based on the leaky integrate-and-fire model, describing neuronal transformation from input signals to a spike train has been proposed for tracking input parameters represented by their mean and fluctuation [11]. In the present paper, we propose to make the estimation more realistic by adopting an LIF model augmented with an adaptive moving threshold. Moreover, because the direct state-space method is computationally infeasible for a data set comprising thousands of spikes, we further develop a practical method for transforming instantaneous firing characteristics back to input parameters. The instantaneous firing characteristics, represented by the firing rate and non-Poisson irregularity, can be estimated using a computationally feasible algorithm. We applied our proposed methods to synthetic data to clarify that they perform well.
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Affiliation(s)
- Hideaki Kim
- NTT Service Evolution Laboratories, NTT Corporation, Yokosuka-shi, Kanagawa, 239-0847, Japan.
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10
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Meng X, Huguet G, Rinzel J. TYPE III EXCITABILITY, SLOPE SENSITIVITY AND COINCIDENCE DETECTION. ACTA ACUST UNITED AC 2012; 32:2729-2757. [PMID: 23667306 DOI: 10.3934/dcds.2012.32.2729] [Citation(s) in RCA: 79] [Impact Index Per Article: 6.6] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/18/2022]
Abstract
Some neurons in the nervous system do not show repetitive firing for steady currents. For time-varying inputs, they fire once if the input rise is fast enough. This property of phasic firing is known as Type III excitability. Type III excitability has been observed in neurons in the auditory brainstem (MSO), which show strong phase-locking and accurate coincidence detection. In this paper, we consider a Hodgkin-Huxley type model (RM03) that is widely-used for phasic MSO neurons and we compare it with a modification of it, showing tonic behavior. We provide insight into the temporal processing of these neuron models by means of developing and analyzing two reduced models that reproduce qualitatively the properties of the exemplar ones. The geometric and mathematical analysis of the reduced models allows us to detect and quantify relevant features for the temporal computation such as nearness to threshold and a temporal integration window. Our results underscore the importance of Type III excitability for precise coincidence detection.
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Affiliation(s)
- Xiangying Meng
- Dynamics and Control, Beihang University, Beijing, China, Center for Neural Science, New York University, USA
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11
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Urdapilleta E. Onset of negative interspike interval correlations in adapting neurons. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2011; 84:041904. [PMID: 22181172 DOI: 10.1103/physreve.84.041904] [Citation(s) in RCA: 7] [Impact Index Per Article: 0.5] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Subscribe] [Scholar Register] [Received: 02/07/2011] [Revised: 07/21/2011] [Indexed: 05/31/2023]
Abstract
Negative serial correlations in single spike trains are an effective method to reduce the variability of spike counts. One of the factors contributing to the development of negative correlations between successive interspike intervals is the presence of adaptation currents. In this work, based on a hidden Markov model and a proper statistical description of conditional responses, we obtain analytically these correlations in an adequate dynamical neuron model resembling adaptation. We derive the serial correlation coefficients for arbitrary lags, under a small adaptation scenario. In this case, the behavior of correlations is universal and depends on the first-order statistical description of an exponentially driven time-inhomogeneous stochastic process.
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Affiliation(s)
- Eugenio Urdapilleta
- División de Física Estadística e Interdisciplinaria & Instituto Balseiro, Centro Atómico Bariloche, Avenida E. Bustillo Km 9.500, S.C. de Bariloche (8400), Río Negro, Argentina.
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12
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Benda J, Maler L, Longtin A. Linear versus nonlinear signal transmission in neuron models with adaptation currents or dynamic thresholds. J Neurophysiol 2011; 104:2806-20. [PMID: 21045213 DOI: 10.1152/jn.00240.2010] [Citation(s) in RCA: 84] [Impact Index Per Article: 6.5] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/22/2022] Open
Abstract
Spike-frequency adaptation is a prominent aspect of neuronal dynamics that shapes a neuron's signal processing properties on timescales ranging from about 10 ms to >1 s. For integrate-and-fire model neurons spike-frequency adaptation is incorporated either as an adaptation current or as a dynamic firing threshold. Whether a physiologically observed adaptation mechanism should be modeled as an adaptation current or a dynamic threshold, however, is not known. Here we show that a dynamic threshold has a divisive effect on the onset f-I curve (the initial maximal firing rate following a step increase in an input current) measured at increasing mean threshold levels, i.e., adaptation states. In contrast, an adaptation current subtractively shifts this f-I curve to higher inputs without affecting its slope. As a consequence, an adaptation current acts essentially linearly, resulting in a high-pass filter component of the neuron's transfer function for current stimuli. With a dynamic threshold, however, the transfer function strongly depends on the input range because of the multiplicative effect on the f-I curves. Simulations of conductance-based spiking models with adaptation currents, such as afterhyperpolarization (AHP)-type, M-type, and sodium-activated potassium currents, do not show the divisive effects of a dynamic threshold, but agree with the properties of integrate-and-fire neurons with adaptation current. Notably, the effects of slow inactivation of sodium currents cannot be reproduced by either model. Our results suggest that, when lateral shifts of the onset f-I curve are seen in response to adapting inputs, adaptation should be modeled with adaptation currents and not with a dynamic threshold. In contrast, when the slope of onset f-I curves depends on the adaptation state, then adaptation should be modeled with a dynamic threshold. Further, the observation of divisively altered onset f-I curves in adapted neurons with notable variability of their spike threshold could hint to yet known biophysical mechanisms directly affecting the threshold.
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Affiliation(s)
- Jan Benda
- Division of Neurobiology, Department Biology II, Ludwig-Maximilians-Universität München, Planegg-Martinsried, Germany.
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Urdapilleta E. Survival probability and first-passage-time statistics of a Wiener process driven by an exponential time-dependent drift. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2011; 83:021102. [PMID: 21405813 DOI: 10.1103/physreve.83.021102] [Citation(s) in RCA: 3] [Impact Index Per Article: 0.2] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 09/04/2010] [Indexed: 05/30/2023]
Abstract
The survival probability and the first-passage-time statistics are important quantities in different fields. The Wiener process is the simplest stochastic process with continuous variables, and important results can be explicitly found from it. The presence of a constant drift does not modify its simplicity; however, when the process has a time-dependent component the analysis becomes difficult. In this work we analyze the statistical properties of the Wiener process with an absorbing boundary, under the effect of an exponential time-dependent drift. Based on the backward Fokker-Planck formalism we set the time-inhomogeneous equation and conditions that rule the diffusion of the corresponding survival probability. We propose as the solution an expansion series in terms of the intensity of the exponential drift, resulting in a set of recurrence equations. We explicitly solve the expansion up to second order and comment on higher-order solutions. The first-passage-time density function arises naturally from the survival probability and preserves the proposed expansion. Explicit results, related properties, and limit behaviors are analyzed and extensively compared to numerical simulations.
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Affiliation(s)
- Eugenio Urdapilleta
- División de Física Estadística e Interdisciplinaria & Instituto Balseiro, Centro Atómico Bariloche, Avenida E. Bustillo Km 9.500, S.C. de Bariloche 8400, Río Negro, Argentina.
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14
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Spectrum of Lyapunov exponents of non-smooth dynamical systems of integrate-and-fire type. J Comput Neurosci 2009; 28:229-45. [DOI: 10.1007/s10827-009-0201-3] [Citation(s) in RCA: 18] [Impact Index Per Article: 1.2] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 09/09/2009] [Revised: 10/31/2009] [Accepted: 11/18/2009] [Indexed: 10/20/2022]
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15
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Vilela RD, Lindner B. Comparative study of different integrate-and-fire neurons: spontaneous activity, dynamical response, and stimulus-induced correlation. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2009; 80:031909. [PMID: 19905148 DOI: 10.1103/physreve.80.031909] [Citation(s) in RCA: 55] [Impact Index Per Article: 3.7] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Subscribe] [Scholar Register] [Received: 01/09/2009] [Revised: 06/27/2009] [Indexed: 05/28/2023]
Abstract
Stochastic integrate-and-fire (IF) neuron models have found widespread applications in computational neuroscience. Here we present results on the white-noise-driven perfect, leaky, and quadratic IF models, focusing on the spectral statistics (power spectra, cross spectra, and coherence functions) in different dynamical regimes (noise-induced and tonic firing regimes with low or moderate noise). We make the models comparable by tuning parameters such that the mean value and the coefficient of variation of the interspike interval (ISI) match for all of them. We find that, under these conditions, the power spectrum under white-noise stimulation is often very similar while the response characteristics, described by the cross spectrum between a fraction of the input noise and the output spike train, can differ drastically. We also investigate how the spike trains of two neurons of the same kind (e.g., two leaky IF neurons) correlate if they share a common noise input. We show that, depending on the dynamical regime, either two quadratic IF models or two leaky IFs are more strongly correlated. Our results suggest that, when choosing among simple IF models for network simulations, the details of the model have a strong effect on correlation and regularity of the output.
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Affiliation(s)
- Rafael D Vilela
- Max-Planck-Institut für Physik Komplexer Systeme, Nöthnitzer Str 38, 01187 Dresden, Germany
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16
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Vilela RD, Lindner B. Are the input parameters of white noise driven integrate and fire neurons uniquely determined by rate and CV? J Theor Biol 2008; 257:90-9. [PMID: 19063904 DOI: 10.1016/j.jtbi.2008.11.004] [Citation(s) in RCA: 39] [Impact Index Per Article: 2.4] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 06/16/2008] [Revised: 10/30/2008] [Accepted: 11/01/2008] [Indexed: 11/12/2022]
Abstract
Integrate and fire (IF) neurons have found widespread applications in computational neuroscience. Particularly important are stochastic versions of these models where the driving consists of a synaptic input modeled as white Gaussian noise with mean mu and noise intensity D. Different IF models have been proposed, the firing statistics of which depends nontrivially on the input parameters mu and D. In order to compare these models among each other, one must first specify the correspondence between their parameters. This can be done by determining which set of parameters (mu,D) of each model is associated with a given set of basic firing statistics as, for instance, the firing rate and the coefficient of variation (CV) of the interspike interval (ISI). However, it is not clear a priori whether for a given firing rate and CV there is only one unique choice of input parameters for each model. Here we review the dependence of rate and CV on input parameters for the perfect, leaky, and quadratic IF neuron models and show analytically that indeed in these three models the firing rate and the CV uniquely determine the input parameters.
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Affiliation(s)
- Rafael D Vilela
- Max-Planck-Institut für Physik komplexer Systeme, Nöthnitzer Str. 38, 01187 Dresden, Germany.
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Badel L, Lefort S, Berger TK, Petersen CCH, Gerstner W, Richardson MJE. Extracting non-linear integrate-and-fire models from experimental data using dynamic I-V curves. BIOLOGICAL CYBERNETICS 2008; 99:361-370. [PMID: 19011924 PMCID: PMC2798053 DOI: 10.1007/s00422-008-0259-4] [Citation(s) in RCA: 40] [Impact Index Per Article: 2.5] [Reference Citation Analysis] [Abstract] [Key Words] [MESH Headings] [Track Full Text] [Subscribe] [Scholar Register] [Received: 07/04/2008] [Accepted: 09/11/2008] [Indexed: 05/27/2023]
Abstract
The dynamic I-V curve method was recently introduced for the efficient experimental generation of reduced neuron models. The method extracts the response properties of a neuron while it is subject to a naturalistic stimulus that mimics in vivo-like fluctuating synaptic drive. The resulting history-dependent, transmembrane current is then projected onto a one-dimensional current-voltage relation that provides the basis for a tractable non-linear integrate-and-fire model. An attractive feature of the method is that it can be used in spike-triggered mode to quantify the distinct patterns of post-spike refractoriness seen in different classes of cortical neuron. The method is first illustrated using a conductance-based model and is then applied experimentally to generate reduced models of cortical layer-5 pyramidal cells and interneurons, in injected-current and injected- conductance protocols. The resulting low-dimensional neuron models-of the refractory exponential integrate-and-fire type-provide highly accurate predictions for spike-times. The method therefore provides a useful tool for the construction of tractable models and rapid experimental classification of cortical neurons.
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Affiliation(s)
- Laurent Badel
- Laboratory of Computational Neuroscience, School of Computer and Communications Sciences and Brain Mind Institute, Ecole Polytechnique Fédérale de Lausanne (EPFL), 1015 Lausanne, Switzerland
| | - Sandrine Lefort
- Laboratory of Sensory Processing, Brain Mind Institute, Ecole Polytechnique Fédérale de Lausanne (EPFL), 1015 Lausanne, Switzerland
| | - Thomas K. Berger
- Laboratory of Neural Microcircuitry, Brain Mind Institute, Ecole Polytechnique Fédérale de Lausanne (EPFL), 1015 Lausanne, Switzerland
| | - Carl C. H. Petersen
- Laboratory of Sensory Processing, Brain Mind Institute, Ecole Polytechnique Fédérale de Lausanne (EPFL), 1015 Lausanne, Switzerland
| | - Wulfram Gerstner
- Laboratory of Computational Neuroscience, School of Computer and Communications Sciences and Brain Mind Institute, Ecole Polytechnique Fédérale de Lausanne (EPFL), 1015 Lausanne, Switzerland
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Badel L, Lefort S, Brette R, Petersen CCH, Gerstner W, Richardson MJE. Dynamic I-V curves are reliable predictors of naturalistic pyramidal-neuron voltage traces. J Neurophysiol 2007; 99:656-66. [PMID: 18057107 DOI: 10.1152/jn.01107.2007] [Citation(s) in RCA: 134] [Impact Index Per Article: 7.9] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/22/2022] Open
Abstract
Neuronal response properties are typically probed by intracellular measurements of current-voltage (I-V) relationships during application of current or voltage steps. Here we demonstrate the measurement of a novel I-V curve measured while the neuron exhibits a fluctuating voltage and emits spikes. This dynamic I-V curve requires only a few tens of seconds of experimental time and so lends itself readily to the rapid classification of cell type, quantification of heterogeneities in cell populations, and generation of reduced analytical models. We apply this technique to layer-5 pyramidal cells and show that their dynamic I-V curve comprises linear and exponential components, providing experimental evidence for a recently proposed theoretical model. The approach also allows us to determine the change of neuronal response properties after a spike, millisecond by millisecond, so that postspike refractoriness of pyramidal cells can be quantified. Observations of I-V curves during and in absence of refractoriness are cast into a model that is used to predict both the subthreshold response and spiking activity of the neuron to novel stimuli. The predictions of the resulting model are in excellent agreement with experimental data and close to the intrinsic neuronal reproducibility to repeated stimuli.
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Affiliation(s)
- Laurent Badel
- Laboratory of Computational Neuroscience, School of Computer and Communication Sciences and Brain Institute, Lausanne, Switzerland
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Daly E, Porporato A. State-dependent fire models and related renewal processes. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2006; 74:041112. [PMID: 17155027 DOI: 10.1103/physreve.74.041112] [Citation(s) in RCA: 3] [Impact Index Per Article: 0.2] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 04/03/2006] [Revised: 06/19/2006] [Indexed: 05/12/2023]
Abstract
We introduce a general class of stochastic processes forced by instantaneous random fires (i.e., jumps) that reset the state variable x to a given value. Since in many physical systems the fire activity is often dependent on the actual value of the state variable, as in the case of natural fires in ecosystems and firing dynamics in neuronal activity, the frequency of fire occurrence is assumed to be state dependent. Such dynamics leads to independent interfire statistics--i.e., to renewal point processes. Various functions relating the frequency of fire occurrence to x(t) are analyzed and compared. The relation between the probabilistic dynamics of x(t) and the interfire statistics is derived and some exact probability distribution of both x(t) and the interfire times are obtained for systems with different degrees of complexity. After studying processes in which the fire activity is coupled only to a deterministic drift, we also analyze processes forced by either additive or multiplicative Gaussian white noise.
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Affiliation(s)
- Edoardo Daly
- Department of Civil and Environmental Engineering & Nicholas School of the Environment and Earth Sciences, Duke University, Durham, North Carolina 27708-0287, USA.
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Burkitt AN. A review of the integrate-and-fire neuron model: I. Homogeneous synaptic input. BIOLOGICAL CYBERNETICS 2006; 95:1-19. [PMID: 16622699 DOI: 10.1007/s00422-006-0068-6] [Citation(s) in RCA: 440] [Impact Index Per Article: 24.4] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Subscribe] [Scholar Register] [Received: 08/02/2005] [Accepted: 03/20/2006] [Indexed: 05/08/2023]
Abstract
The integrate-and-fire neuron model is one of the most widely used models for analyzing the behavior of neural systems. It describes the membrane potential of a neuron in terms of the synaptic inputs and the injected current that it receives. An action potential (spike) is generated when the membrane potential reaches a threshold, but the actual changes associated with the membrane voltage and conductances driving the action potential do not form part of the model. The synaptic inputs to the neuron are considered to be stochastic and are described as a temporally homogeneous Poisson process. Methods and results for both current synapses and conductance synapses are examined in the diffusion approximation, where the individual contributions to the postsynaptic potential are small. The focus of this review is upon the mathematical techniques that give the time distribution of output spikes, namely stochastic differential equations and the Fokker-Planck equation. The integrate-and-fire neuron model has become established as a canonical model for the description of spiking neurons because it is capable of being analyzed mathematically while at the same time being sufficiently complex to capture many of the essential features of neural processing. A number of variations of the model are discussed, together with the relationship with the Hodgkin-Huxley neuron model and the comparison with electrophysiological data. A brief overview is given of two issues in neural information processing that the integrate-and-fire neuron model has contributed to - the irregular nature of spiking in cortical neurons and neural gain modulation.
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Affiliation(s)
- A N Burkitt
- The Bionic Ear Institute, 384-388 Albert Street, East Melbourne, VIC, 3002, Australia.
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