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Shomali SR, Rasuli SN, Ahmadabadi MN, Shimazaki H. Uncovering hidden network architecture from spiking activities using an exact statistical input-output relation of neurons. Commun Biol 2023; 6:169. [PMID: 36792689 PMCID: PMC9932086 DOI: 10.1038/s42003-023-04511-z] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 02/03/2022] [Accepted: 01/20/2023] [Indexed: 02/17/2023] Open
Abstract
Identifying network architecture from observed neural activities is crucial in neuroscience studies. A key requirement is knowledge of the statistical input-output relation of single neurons in vivo. By utilizing an exact analytical solution of the spike-timing for leaky integrate-and-fire neurons under noisy inputs balanced near the threshold, we construct a framework that links synaptic type, strength, and spiking nonlinearity with the statistics of neuronal population activity. The framework explains structured pairwise and higher-order interactions of neurons receiving common inputs under different architectures. We compared the theoretical predictions with the activity of monkey and mouse V1 neurons and found that excitatory inputs given to pairs explained the observed sparse activity characterized by strong negative triple-wise interactions, thereby ruling out the alternative explanation by shared inhibition. Moreover, we showed that the strong interactions are a signature of excitatory rather than inhibitory inputs whenever the spontaneous rate is low. We present a guide map of neural interactions that help researchers to specify the hidden neuronal motifs underlying observed interactions found in empirical data.
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Affiliation(s)
- Safura Rashid Shomali
- School of Cognitive Sciences, Institute for Research in Fundamental Sciences (IPM), Tehran, 19395-5746, Iran.
| | - Seyyed Nader Rasuli
- grid.418744.a0000 0000 8841 7951School of Physics, Institute for Research in Fundamental Sciences (IPM), Tehran, 19395-5531 Iran ,grid.411872.90000 0001 2087 2250Department of Physics, University of Guilan, Rasht, 41335-1914 Iran
| | - Majid Nili Ahmadabadi
- grid.46072.370000 0004 0612 7950Control and Intelligent Processing Center of Excellence, School of Electrical and Computer Engineering, College of Engineering, University of Tehran, Tehran, 14395-515 Iran
| | - Hideaki Shimazaki
- Graduate School of Informatics, Kyoto University, Kyoto, 606-8501, Japan. .,Center for Human Nature, Artificial Intelligence, and Neuroscience (CHAIN), Hokkaido University, Hokkaido, 060-0812, Japan.
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2
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Schwalger T. Mapping input noise to escape noise in integrate-and-fire neurons: a level-crossing approach. BIOLOGICAL CYBERNETICS 2021; 115:539-562. [PMID: 34668051 PMCID: PMC8551127 DOI: 10.1007/s00422-021-00899-1] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Key Words] [MESH Headings] [Grants] [Track Full Text] [Figures] [Subscribe] [Scholar Register] [Received: 07/17/2021] [Accepted: 09/27/2021] [Indexed: 06/13/2023]
Abstract
Noise in spiking neurons is commonly modeled by a noisy input current or by generating output spikes stochastically with a voltage-dependent hazard rate ("escape noise"). While input noise lends itself to modeling biophysical noise processes, the phenomenological escape noise is mathematically more tractable. Using the level-crossing theory for differentiable Gaussian processes, we derive an approximate mapping between colored input noise and escape noise in leaky integrate-and-fire neurons. This mapping requires the first-passage-time (FPT) density of an overdamped Brownian particle driven by colored noise with respect to an arbitrarily moving boundary. Starting from the Wiener-Rice series for the FPT density, we apply the second-order decoupling approximation of Stratonovich to the case of moving boundaries and derive a simplified hazard-rate representation that is local in time and numerically efficient. This simplification requires the calculation of the non-stationary auto-correlation function of the level-crossing process: For exponentially correlated input noise (Ornstein-Uhlenbeck process), we obtain an exact formula for the zero-lag auto-correlation as a function of noise parameters, mean membrane potential and its speed, as well as an exponential approximation of the full auto-correlation function. The theory well predicts the FPT and interspike interval densities as well as the population activities obtained from simulations with colored input noise and time-dependent stimulus or boundary. The agreement with simulations is strongly enhanced across the sub- and suprathreshold firing regime compared to a first-order decoupling approximation that neglects correlations between level crossings. The second-order approximation also improves upon a previously proposed theory in the subthreshold regime. Depending on a simplicity-accuracy trade-off, all considered approximations represent useful mappings from colored input noise to escape noise, enabling progress in the theory of neuronal population dynamics.
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Affiliation(s)
- Tilo Schwalger
- Institute of Mathematics, Technical University Berlin, 10623, Berlin, Germany.
- Bernstein Center for Computational Neuroscience Berlin, 10115, Berlin, Germany.
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3
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Zhao Y, Gu C, Yang H. Visibility-graphlet approach to the output series of a Hodgkin-Huxley neuron. CHAOS (WOODBURY, N.Y.) 2021; 31:043102. [PMID: 34251267 DOI: 10.1063/5.0018359] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Subscribe] [Scholar Register] [Received: 06/14/2020] [Accepted: 03/10/2021] [Indexed: 06/13/2023]
Abstract
The output signals of neurons that are exposed to external stimuli are of great importance for brain functionality. Traditional time-series analysis methods have provided encouraging results; however, the associated patterns and their correlations in the output signals of neurons are masked by statistical procedures. Here, graphlets are employed to extract the local temporal patterns and the transitions between them from the output signals when neurons are exposed to external stimuli with selected stimulating periods. A transition network is defined where the node is the graphlet and the direct link is the transition between two successive graphlets. The transition-network structure is affected by the simulating periods. When the stimulating period moves close to an integer multiple of the neuronal intrinsic period, only the backbone or core survives, while the other linkages disappear. Interestingly, the size of the backbone (number of nodes) equals the multiple. The transition-network structure is conservative within each stimulating region, which is defined as the range between two successive integer multiples. Nevertheless, the backbone or detailed structure is significantly altered between different stimulating regions. This alternation is induced primarily from a total of 12 active linkages. Hence, the transition network shows the structure of cross correlations in the output time-series for a single neuron.
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Affiliation(s)
- Yuanying Zhao
- Business School, University of Shanghai for Science and Technology, Shanghai 200093, People's Republic of China
| | - Changgui Gu
- Business School, University of Shanghai for Science and Technology, Shanghai 200093, People's Republic of China
| | - Huijie Yang
- Business School, University of Shanghai for Science and Technology, Shanghai 200093, People's Republic of China
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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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New Type of Spectral Nonlinear Resonance Enhances Identification of Weak Signals. Sci Rep 2019; 9:14125. [PMID: 31575962 PMCID: PMC6773744 DOI: 10.1038/s41598-019-50767-z] [Citation(s) in RCA: 3] [Impact Index Per Article: 0.6] [Reference Citation Analysis] [Abstract] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 07/04/2019] [Accepted: 09/18/2019] [Indexed: 11/09/2022] Open
Abstract
Some nonlinear systems possess innate capabilities of enhancing weak signal transmissions through a unique process called Stochastic Resonance (SR). However, existing SR mechanism suffers limited signal enhancement from inappropriate entraining signals. Here we propose a new and effective implementation, resulting in a new type of spectral resonance similar to SR but capable of achieving orders of magnitude higher signal enhancement than previously reported. By employing entraining frequency in the range of the weak signal, strong spectral resonances can be induced to facilitate nonlinear modulations and intermodulations, thereby strengthening the weak signal. The underlying physical mechanism governing the behavior of spectral resonances is examined, revealing the inherent advantages of the proposed spectral resonances over the existing implementation of SR. Wide range of parameters have been found for the optimal enhancement of any given weak signal and an analytical method is established to estimate these required parameters. A reliable algorithm is also developed for the identifications of weak signals using signal processing techniques. The present work can significantly improve existing SR performances and can have profound practical applications where SR is currently employed for its inherent technological advantages.
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Multiplicative noise is beneficial for the transmission of sensory signals in simple neuron models. Biosystems 2019; 178:25-31. [PMID: 30735693 DOI: 10.1016/j.biosystems.2019.02.002] [Citation(s) in RCA: 9] [Impact Index Per Article: 1.8] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 10/24/2018] [Revised: 01/27/2019] [Accepted: 02/04/2019] [Indexed: 11/23/2022]
Abstract
We study simple integrate-and-fire type models with multiplicative noise and consider the transmission of a weak and slow signal, i.e. a signal that evokes a small modulation of the instantaneous firing rate on time scales that are much larger than the membrane time scale and the mean interspike interval. The specific question of interest is whether and how the state-dependence of the noise can be optimized with respect to information transmission. First, in a simple model in which the noise intensity varies linearly with the state variable, we show analytically that multiplicative fluctuations may benefit the signal transfer and we elucidate the mechanism for this improvement. In a conductance-based integrate-and-fire model with synaptically filtered shot-noise input, we show by means of extended numerical simulations that also in a biophysically more relevant situation, multiplicative noise can enhance the signal-to-noise ratio. Our results shed light on a so far unexplored aspect of stochastic signal transmission in neural systems.
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Shomali SR, Ahmadabadi MN, Shimazaki H, Rasuli SN. How does transient signaling input affect the spike timing of postsynaptic neuron near the threshold regime: an analytical study. J Comput Neurosci 2017; 44:147-171. [PMID: 29192377 PMCID: PMC5851711 DOI: 10.1007/s10827-017-0664-6] [Citation(s) in RCA: 2] [Impact Index Per Article: 0.3] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 11/02/2016] [Revised: 07/14/2017] [Accepted: 09/11/2017] [Indexed: 11/05/2022]
Abstract
The noisy threshold regime, where even a small set of presynaptic neurons can significantly affect postsynaptic spike-timing, is suggested as a key requisite for computation in neurons with high variability. It also has been proposed that signals under the noisy conditions are successfully transferred by a few strong synapses and/or by an assembly of nearly synchronous synaptic activities. We analytically investigate the impact of a transient signaling input on a leaky integrate-and-fire postsynaptic neuron that receives background noise near the threshold regime. The signaling input models a single strong synapse or a set of synchronous synapses, while the background noise represents a lot of weak synapses. We find an analytic solution that explains how the first-passage time (ISI) density is changed by transient signaling input. The analysis allows us to connect properties of the signaling input like spike timing and amplitude with postsynaptic first-passage time density in a noisy environment. Based on the analytic solution, we calculate the Fisher information with respect to the signaling input’s amplitude. For a wide range of amplitudes, we observe a non-monotonic behavior for the Fisher information as a function of background noise. Moreover, Fisher information non-trivially depends on the signaling input’s amplitude; changing the amplitude, we observe one maximum in the high level of the background noise. The single maximum splits into two maximums in the low noise regime. This finding demonstrates the benefit of the analytic solution in investigating signal transfer by neurons.
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Affiliation(s)
- Safura Rashid Shomali
- School of Cognitive Sciences, Institute for Research in Fundamental Sciences (IPM), P.O. Box 19395-5746 (1954851167), Tehran, Iran.
| | - Majid Nili Ahmadabadi
- Control and Intelligent Processing Center of Excellence, School of Electrical and Computer Engineering, College of Engineering, University of Tehran, Tehran, 14395-515, Iran
| | - Hideaki Shimazaki
- Graduate School of Informatics, Kyoto University, Yoshida-honmachi, Sakyo-ku, Kyoto, 606-8501, Japan.,Honda Research Institute Japan, Honcho 8-1, Wako-shi, Saitama, 351-0188, Japan
| | - Seyyed Nader Rasuli
- Department of Physics, University of Guilan, Rasht, 41335-1914, Iran.,School of Physics, Institute for Research in Fundamental Sciences (IPM), P.O. Box 19395-5531, Tehran, Iran
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8
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Addition of visual noise boosts evoked potential-based brain-computer interface. Sci Rep 2014; 4:4953. [PMID: 24828128 PMCID: PMC4021798 DOI: 10.1038/srep04953] [Citation(s) in RCA: 18] [Impact Index Per Article: 1.8] [Reference Citation Analysis] [Abstract] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 10/31/2013] [Accepted: 04/17/2014] [Indexed: 11/08/2022] Open
Abstract
Although noise has a proven beneficial role in brain functions, there have not been any attempts on the dedication of stochastic resonance effect in neural engineering applications, especially in researches of brain-computer interfaces (BCIs). In our study, a steady-state motion visual evoked potential (SSMVEP)-based BCI with periodic visual stimulation plus moderate spatiotemporal noise can achieve better offline and online performance due to enhancement of periodic components in brain responses, which was accompanied by suppression of high harmonics. Offline results behaved with a bell-shaped resonance-like functionality and 7–36% online performance improvements can be achieved when identical visual noise was adopted for different stimulation frequencies. Using neural encoding modeling, these phenomena can be explained as noise-induced input-output synchronization in human sensory systems which commonly possess a low-pass property. Our work demonstrated that noise could boost BCIs in addressing human needs.
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Iolov A, Ditlevsen S, Longtin A. Fokker-Planck and Fortet Equation-Based Parameter Estimation for a Leaky Integrate-and-Fire Model with Sinusoidal and Stochastic Forcing. JOURNAL OF MATHEMATICAL NEUROSCIENCE 2014; 4:4. [PMID: 24742022 PMCID: PMC4234988 DOI: 10.1186/2190-8567-4-4] [Citation(s) in RCA: 4] [Impact Index Per Article: 0.4] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Figures] [Subscribe] [Scholar Register] [Received: 12/30/2012] [Accepted: 06/06/2013] [Indexed: 06/03/2023]
Abstract
UNLABELLED 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 Fokker-Planck 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-fireISI: Interspike intervalSDE: Stochastic differential equationPDE: Partial differential equation.
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Affiliation(s)
- Alexandre Iolov
- Department of Mathematics and Statistics, University of Ottawa, Ottawa, Canada
- Department of Mathematical Sciences, University of Copenhagen, Copenhagen, Denmark
| | - Susanne Ditlevsen
- Department of Mathematical Sciences, University of Copenhagen, Copenhagen, Denmark
| | - André Longtin
- Department of Physics and Center for Neural Dynamics, University of Ottawa, Ottawa, Canada
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10
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Sirovich R, Sacerdote L, Villa AEP. Cooperative behavior in a jump diffusion model for a simple network of spiking neurons. MATHEMATICAL BIOSCIENCES AND ENGINEERING : MBE 2014; 11:385-401. [PMID: 24245723 DOI: 10.3934/mbe.2014.11.385] [Citation(s) in RCA: 10] [Impact Index Per Article: 1.0] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 06/02/2023]
Abstract
The distribution of time intervals between successive spikes generated by a neuronal cell --the interspike intervals (ISI)-- may reveal interesting features of the underlying dynamics. In this study we analyze the ISI sequence --the spike train-- generated by a simple network of neurons whose output activity is modeled by a jump-diffusion process. We prove that, when specific ranges of the involved parameters are chosen, it is possible to observe multimodal ISI distributions which reveal that the modeled network fires with more than one single preferred time interval. Furthermore, the system exhibits resonance behavior, with modulation of the spike timings by the noise intensity. We also show that inhibition helps the signal transmission between the units of the simple network.
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Affiliation(s)
- Roberta Sirovich
- Department of Mathematics "G. Peano", University of Torino, Via Carlo Alberto 10, 10123 Torino, Italy.
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11
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Bauermeister C, Schwalger T, Russell DF, Neiman AB, Lindner B. Characteristic effects of stochastic oscillatory forcing on neural firing: analytical theory and comparison to paddlefish electroreceptor data. PLoS Comput Biol 2013; 9:e1003170. [PMID: 23966844 PMCID: PMC3744407 DOI: 10.1371/journal.pcbi.1003170] [Citation(s) in RCA: 15] [Impact Index Per Article: 1.4] [Reference Citation Analysis] [Abstract] [MESH Headings] [Grants] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 12/13/2012] [Accepted: 06/21/2013] [Indexed: 11/18/2022] Open
Abstract
Stochastic signals with pronounced oscillatory components are frequently encountered in neural systems. Input currents to a neuron in the form of stochastic oscillations could be of exogenous origin, e.g. sensory input or synaptic input from a network rhythm. They shape spike firing statistics in a characteristic way, which we explore theoretically in this report. We consider a perfect integrate-and-fire neuron that is stimulated by a constant base current (to drive regular spontaneous firing), along with Gaussian narrow-band noise (a simple example of stochastic oscillations), and a broadband noise. We derive expressions for the nth-order interval distribution, its variance, and the serial correlation coefficients of the interspike intervals (ISIs) and confirm these analytical results by computer simulations. The theory is then applied to experimental data from electroreceptors of paddlefish, which have two distinct types of internal noisy oscillators, one forcing the other. The theory provides an analytical description of their afferent spiking statistics during spontaneous firing, and replicates a pronounced dependence of ISI serial correlation coefficients on the relative frequency of the driving oscillations, and furthermore allows extraction of certain parameters of the intrinsic oscillators embedded in these electroreceptors.
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Affiliation(s)
| | - Tilo Schwalger
- Max-Planck-Institute for the Physics of Complex Systems, Dresden, Germany
- Bernstein Center for Computational Neuroscience and Physics Department of Humboldt University, Berlin, Germany
| | - David F. Russell
- Department of Biological Sciences and Neuroscience Program, Ohio University, Athens, Ohio, United States of America
| | - Alexander B. Neiman
- Department of Physics and Astronomy and Neuroscience Program, Ohio University, Athens, Ohio, United States of America
| | - Benjamin Lindner
- Max-Planck-Institute for the Physics of Complex Systems, Dresden, Germany
- Bernstein Center for Computational Neuroscience and Physics Department of Humboldt University, Berlin, Germany
- * E-mail:
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12
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Zare M, Grigolini P. Cooperation in neural systems: bridging complexity and periodicity. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2012; 86:051918. [PMID: 23214825 DOI: 10.1103/physreve.86.051918] [Citation(s) in RCA: 3] [Impact Index Per Article: 0.3] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Subscribe] [Scholar Register] [Received: 07/31/2012] [Revised: 10/22/2012] [Indexed: 06/01/2023]
Abstract
Inverse power law distributions are generally interpreted as a manifestation of complexity, and waiting time distributions with power index μ<2 reflect the occurrence of ergodicity-breaking renewal events. In this paper we show how to combine these properties with the apparently foreign clocklike nature of biological processes. We use a two-dimensional regular network of leaky integrate-and-fire neurons, each of which is linked to its four nearest neighbors, to show that both complexity and periodicity are generated by locality breakdown: Links of increasing strength have the effect of turning local interactions into long-range interactions, thereby generating time complexity followed by time periodicity. Increasing the density of neuron firings reduces the influence of periodicity, thus creating a cooperation-induced renewal condition that is distinctly non-Poissonian.
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Affiliation(s)
- Marzieh Zare
- Center for Nonlinear Science, University of North Texas, PO Box 311427, Denton, Texas 76203-1427, USA
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13
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Abstract
We derive a model of a neuron’s interspike interval probability density through analysis of the first passage problem. The fit of our expression to retinal ganglion cell laboratory data extracts three physiologically relevant parameters, with which our model yields input-output features that conform to laboratory results. Preliminary analysis suggests that under common circumstances, local circuitry readjusts these parameters with changes in firing rate and so endeavors to faithfully replicate an input signal. Further results suggest that the so-called principle of sloppy workmanship also plays a role in evolution’s choice of these parameters.
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Affiliation(s)
- Lawrence Sirovich
- Laboratory of Applied Mathematics, Mount Sinai School of Medicine, New York, NY 10029, U.S.A
| | - Bruce Knight
- Center for Studies in Physics and Biology, Rockefeller University, New York, NY 10065, U.S.A
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14
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Liang LZJ, Lemmens D, Tempere J. Path integral approach to the pricing of timer options with the Duru-Kleinert time transformation. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2011; 83:056112. [PMID: 21728610 DOI: 10.1103/physreve.83.056112] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 01/19/2011] [Indexed: 05/31/2023]
Abstract
In this paper, a time substitution as used by Duru and Kleinert in their treatment of the hydrogen atom with path integrals is performed to price timer options under stochastic volatility models. We present general pricing formulas for both the perpetual timer call options and the finite time-horizon timer call options. These general results allow us to find closed-form pricing formulas for both the perpetual and the finite time-horizon timer options under the 3/2 stochastic volatility model as well as under the Heston stochastic volatility model. For the treatment of timer options under the 3/2 model we will rely on the path integral for the Morse potential, with the Heston model we will rely on the Kratzer potential.
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Affiliation(s)
- L Z J Liang
- TQC, Universiteit Antwerpen, Universiteitsplein 1, B-2610 Antwerpen, Belgium
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15
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Mejias JF, Torres JJ. Emergence of resonances in neural systems: the interplay between adaptive threshold and short-term synaptic plasticity. PLoS One 2011; 6:e17255. [PMID: 21408148 PMCID: PMC3050837 DOI: 10.1371/journal.pone.0017255] [Citation(s) in RCA: 24] [Impact Index Per Article: 1.8] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 11/01/2010] [Accepted: 01/25/2011] [Indexed: 11/18/2022] Open
Abstract
In this work we study the detection of weak stimuli by spiking (integrate-and-fire) neurons in the presence of certain level of noisy background neural activity. Our study has focused in the realistic assumption that the synapses in the network present activity-dependent processes, such as short-term synaptic depression and facilitation. Employing mean-field techniques as well as numerical simulations, we found that there are two possible noise levels which optimize signal transmission. This new finding is in contrast with the classical theory of stochastic resonance which is able to predict only one optimal level of noise. We found that the complex interplay between adaptive neuron threshold and activity-dependent synaptic mechanisms is responsible for this new phenomenology. Our main results are confirmed by employing a more realistic FitzHugh-Nagumo neuron model, which displays threshold variability, as well as by considering more realistic stochastic synaptic models and realistic signals such as poissonian spike trains.
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Affiliation(s)
- Jorge F Mejias
- Center for Neural Dynamics, University of Ottawa, Ottawa, Ontario, Canada.
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16
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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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17
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Pawlas Z, Lansky P. Distribution of interspike intervals estimated from multiple spike trains observed in a short time window. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2011; 83:011910. [PMID: 21405716 DOI: 10.1103/physreve.83.011910] [Citation(s) in RCA: 3] [Impact Index Per Article: 0.2] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Subscribe] [Scholar Register] [Received: 07/26/2010] [Indexed: 05/30/2023]
Abstract
Several nonparametric estimators of the probability distribution of interspike intervals are introduced. The methods are suitable for simultaneous spike trains observed in a time window of length comparable with the mean interspike interval. This reflects the situation in which a high number of input spike trains converge to a single cortical neuron that has to react in a relatively short time. The simulation study is performed to compare the estimators. For that purpose, several types of stationary point processes are considered as the models of neuronal activity. The methods permit one to estimate the distribution of interspike intervals even if practically none of them are observed. The Kaplan-Meier estimator seems to be the most flexible and reliable among all studied methods, but no direct conclusions as to how real neurons work can be deduced from it.
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Affiliation(s)
- Zbyněk Pawlas
- Department of Probability and Mathematical Statistics, Faculty of Mathematics and Physics, Charles University, Prague, Czech Republic.
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18
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Integrate-and-fire models of insolation-driven entrainment of broadcast spawning in corals. THEOR ECOL-NETH 2010. [DOI: 10.1007/s12080-010-0075-z] [Citation(s) in RCA: 4] [Impact Index Per Article: 0.3] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 10/19/2022]
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Bibbona E, Lansky P, Sirovich R. Estimating input parameters from intracellular recordings in the Feller neuronal model. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2010; 81:031916. [PMID: 20365779 DOI: 10.1103/physreve.81.031916] [Citation(s) in RCA: 3] [Impact Index Per Article: 0.2] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Subscribe] [Scholar Register] [Received: 11/10/2009] [Indexed: 05/29/2023]
Abstract
We study the estimation of the input parameters in a Feller neuronal model from a trajectory of the membrane potential sampled at discrete times. These input parameters are identified with the drift and the infinitesimal variance of the underlying stochastic diffusion process with multiplicative noise. The state space of the process is restricted from below by an inaccessible boundary. Further, the model is characterized by the presence of an absorbing threshold, the first hitting of which determines the length of each trajectory and which constrains the state space from above. We compare, both in the presence and in the absence of the absorbing threshold, the efficiency of different known estimators. In addition, we propose an estimator for the drift term, which is proved to be more efficient than the others, at least in the explored range of the parameters. The presence of the threshold makes the estimates of the drift term biased, and two methods to correct it are proposed.
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Affiliation(s)
- Enrico Bibbona
- Department of Mathematics G Peano, University of Torino, Via Carlo Alberto 10, 10123 Torino, Italy.
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20
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Urdapilleta E, Samengo I. Quasistatic approximation of the interspike interval distribution of neurons driven by time-dependent inputs. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2009; 80:011915. [PMID: 19658737 DOI: 10.1103/physreve.80.011915] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Subscribe] [Scholar Register] [Received: 02/24/2009] [Revised: 04/23/2009] [Indexed: 05/28/2023]
Abstract
Variability in neural responses is a ubiquitous phenomenon in neurons, usually modeled with stochastic differential equations. In particular, stochastic integrate-and-fire models are widely used to simplify theoretical studies. The statistical properties of the generated spikes depend on the stimulating input current. Given that real sensory neurons are driven by time-dependent signals, here we study how the interspike interval distribution of integrate-and-fire neurons depends on the evolution of the stimulus in a quasistatic limit. We obtain a closed-form expression for this distribution, and we compare it to the one obtained with numerical simulations for several time-dependent currents. For slow inputs, the quasistatic distribution provides a very good description of the data. The results obtained for the integrate-and-fire model can be extended to other nonautonomous stochastic systems where the first passage time problem has an explicit solution.
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Affiliation(s)
- Eugenio Urdapilleta
- División de Física Estadística e Interdisciplinaria and Instituto Balseiro, Centro Atómico Bariloche, Av. E. Bustillo Km 9.500, S. C. de Bariloche, 8400 Río Negro, Argentina.
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21
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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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22
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Lansky P, Ditlevsen S. A review of the methods for signal estimation in stochastic diffusion leaky integrate-and-fire neuronal models. BIOLOGICAL CYBERNETICS 2008; 99:253-262. [PMID: 18496710 DOI: 10.1007/s00422-008-0237-x] [Citation(s) in RCA: 32] [Impact Index Per Article: 2.0] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Subscribe] [Scholar Register] [Received: 01/29/2008] [Accepted: 04/24/2008] [Indexed: 05/26/2023]
Abstract
Parameters in diffusion neuronal models are divided into two groups; intrinsic and input parameters. Intrinsic parameters are related to the properties of the neuronal membrane and are assumed to be known throughout the paper. Input parameters characterize processes generated outside the neuron and methods for their estimation are reviewed here. Two examples of the diffusion neuronal model, which are based on the integrate-and-fire concept, are investigated--the Ornstein--Uhlenbeck model as the most common one and the Feller model as an illustration of state-dependent behavior in modeling the neuronal input. Two types of experimental data are assumed-intracellular describing the membrane trajectories and extracellular resulting in knowledge of the interspike intervals. The literature on estimation from the trajectories of the diffusion process is extensive and thus the stress in this review is set on the inference made from the interspike intervals.
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Affiliation(s)
- Petr Lansky
- Institute of Physiology, Academy of Sciences of the Czech Republic, Prague, Czech Republic
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23
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Buonocore A, Caputo L, Pirozzi E. On the evaluation of firing densities for periodically driven neuron models. Math Biosci 2008; 214:122-33. [PMID: 18374954 DOI: 10.1016/j.mbs.2008.02.003] [Citation(s) in RCA: 4] [Impact Index Per Article: 0.3] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 12/29/2007] [Revised: 02/01/2008] [Accepted: 02/09/2008] [Indexed: 10/22/2022]
Abstract
The leaky integrate-and-fire model for neuronal spiking events driven by a periodic stimulus is studied by using the Fokker-Planck formulation. To this purpose, an essential use is made of the asymptotic behavior of the first-passage-time probability density function of a time homogeneous diffusion process through an asymptotically periodic threshold. Numerical comparisons with some recently published results derived by a different approach are performed. Use of a new asymptotic approximation is then made in order to design a numerical algorithm of predictor-corrector type to solve the integral equation in the unknown first-passage-time probability density function. Such algorithm, characterized by a reduced (linear) computation time, is seen to provide a high computation accuracy. Finally, it is shown that such an approach yields excellent approximations to the firing probability density function for a wide range of parameters, including the case of high stimulus frequencies.
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Affiliation(s)
- Aniello Buonocore
- Dipartimento di Matematica e Applicazioni, Università di Napoli Federico II, Via Cintia, 80126 Napoli, Italy.
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24
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Schwalger T, Schimansky-Geier L. Interspike interval statistics of a leaky integrate-and-fire neuron driven by Gaussian noise with large correlation times. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2008; 77:031914. [PMID: 18517429 DOI: 10.1103/physreve.77.031914] [Citation(s) in RCA: 13] [Impact Index Per Article: 0.8] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Subscribe] [Scholar Register] [Received: 10/25/2007] [Revised: 12/19/2007] [Indexed: 05/26/2023]
Abstract
We analytically investigate the interspike interval (ISI) density, the Fano factor, and the coefficient of variation of a leaky integrate-and-fire neuron model driven by exponentially correlated Gaussian noise with a large correlation time tau . We find a burstinglike behavior of the spike train, which is revealed by a dominant peak of the ISI density at small intraburst intervals and a slow power-law decay of long interburst intervals. The large, power-law distributed ISIs give rise to a coefficient of variation which diverges as square root [tau] . This leads to the paradoxical effect that ISI correlations, as expressed by the serial correlation coefficient, vanish for large correlation times. This is in contrast to findings of previous works on a simpler neuron model where the effect of noise correlations appeared in higher-order statistical measures.
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Affiliation(s)
- Tilo Schwalger
- Humboldt-Universität Berlin, Newtonstrasse 15, D-12489 Berlin, Germany and RIKEN Brain Science Institute, 2-1 Hirosawa, Wako, Saitama 351-0198, Japan
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25
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26
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Ditlevsen S, Lansky P. Parameters of stochastic diffusion processes estimated from observations of first-hitting times: application to the leaky integrate-and-fire neuronal model. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2007; 76:041906. [PMID: 17995025 DOI: 10.1103/physreve.76.041906] [Citation(s) in RCA: 9] [Impact Index Per Article: 0.5] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Subscribe] [Scholar Register] [Received: 01/17/2007] [Indexed: 05/25/2023]
Abstract
A theoretical model has to stand the test against the real world to be of any practical use. The first step is to identify parameters in the model estimated from experimental data. In many applications where renewal point data are available, models of first-hitting times of underlying diffusion processes arise. Despite the seemingly simplicity of the model, the problem of how to estimate parameters of the underlying stochastic process has resisted solution. The few attempts have either been unreliable, difficult to implement, or only valid in subsets of the relevant parameter space. Here we present an estimation method that overcomes these difficulties, is computationally easy and fast to implement, and also works surprisingly well on small data sets. The method is illustrated on simulated and experimental data. Two common neuronal models--the Ornstein-Uhlenbeck and Feller models--are investigated.
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Affiliation(s)
- Susanne Ditlevsen
- Department of Biostatistics, University of Copenhagen, Øster Farimagsgade 5, 1014 Copenhagen K, Denmark.
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27
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Lansky P, Sacerdote L, Zucca C. Optimum signal in a diffusion leaky integrate-and-fire neuronal model. Math Biosci 2007; 207:261-74. [PMID: 17070558 DOI: 10.1016/j.mbs.2006.08.027] [Citation(s) in RCA: 9] [Impact Index Per Article: 0.5] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 05/23/2006] [Revised: 08/14/2006] [Accepted: 08/30/2006] [Indexed: 11/21/2022]
Abstract
An optimum signal in the Ornstein-Uhlenbeck neuronal model is determined on the basis of interspike interval data. Two criteria are proposed for this purpose. The first, the classical one, is based on searching for maxima of the slope of the frequency transfer function. The second one uses maximum of the Fisher information, which is, under certain conditions, the inverse variance of the best possible estimator. The Fisher information is further normalized with respect to the time required to make the observation on which the signal estimation is performed. Three variants of the model are investigated. Beside the basic one, we use the version obtained by inclusion of the refractory period. Finally, we investigate such a version of the model in which signal and the input parameter of the model are in a nonlinear relationship. The results show that despite qualitative similarity between the criteria, there is substantial quantitative difference. As a common feature, we found that in the Ornstein-Uhlenbeck model with increasing noise the optimum signal decreases and the coding range gets broader.
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Affiliation(s)
- Petr Lansky
- Institute of Physiology, Academy of Sciences of Czech Republic, Videnska 1083, 142 20 Prague 4, Czech Republic.
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28
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Barbi M, Di Garbo A, Barbi F. Stochastic resonance in two simple compare-and-fire models. Biosystems 2006; 89:58-62. [PMID: 17178434 DOI: 10.1016/j.biosystems.2006.05.011] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.1] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 12/15/2005] [Accepted: 05/15/2006] [Indexed: 10/23/2022]
Abstract
Two neural models are analysed and shown to exhibit the stochastic resonance effect. Namely, they respond to an underthreshold sinusoidal signal with an output signal whose signal-to-noise ratio (SNR) firstly increases then decreases as the intensity of noise affecting the system increases. The resonance curves are determined, analytically for the first and simplest model and by a synthetic method for the second one, and the respective resonant behaviours are illustrated and interpreted.
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Affiliation(s)
- Michele Barbi
- Istituto di Biofisica del CNR, Via Moruzzi 1, 56124 Pisa, Italy.
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29
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Two-compartment stochastic model of a neuron with periodic input. ACTA ACUST UNITED AC 2006. [DOI: 10.1007/bfb0098179] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register]
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30
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Burkitt AN. A review of the integrate-and-fire neuron model: II. Inhomogeneous synaptic input and network properties. BIOLOGICAL CYBERNETICS 2006; 95:97-112. [PMID: 16821035 DOI: 10.1007/s00422-006-0082-8] [Citation(s) in RCA: 130] [Impact Index Per Article: 7.2] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Subscribe] [Scholar Register] [Received: 08/02/2005] [Accepted: 05/29/2006] [Indexed: 05/08/2023]
Abstract
The integrate-and-fire neuron model describes the state of a neuron in terms of its membrane potential, which is determined by the synaptic inputs and the injected current that the neuron receives. When the membrane potential reaches a threshold, an action potential (spike) is generated. This review considers the model in which the synaptic input varies periodically and is described by an inhomogeneous Poisson process, with both current and conductance synapses. The focus is on the mathematical methods that allow the output spike distribution to be analyzed, including first passage time methods and the Fokker-Planck equation. Recent interest in the response of neurons to periodic input has in part arisen from the study of stochastic resonance, which is the noise-induced enhancement of the signal-to-noise ratio. Networks of integrate-and-fire neurons behave in a wide variety of ways and have been used to model a variety of neural, physiological, and psychological phenomena. The properties of the integrate-and-fire neuron model with synaptic input described as a temporally homogeneous Poisson process are reviewed in an accompanying paper (Burkitt in Biol Cybern, 2006).
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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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31
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Reinker S, Li YX, Kuske R. Noise-Induced Coherence and Network Oscillations in a Reduced Bursting Model. Bull Math Biol 2006; 68:1401-27. [PMID: 17149822 DOI: 10.1007/s11538-006-9089-5] [Citation(s) in RCA: 4] [Impact Index Per Article: 0.2] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 10/24/2022]
Abstract
The dynamics of the Hindmarsh-Rose (HR) model of bursting thalamic neurons is reduced to a system of two linear differential equations that retains the subthreshold resonance properties of the HR model. Introducing a reset mechanism after a threshold crossing, we turn this system into a resonant integrate-and-fire (RIF) model. Using Monte-Carlo simulations and mathematical analysis, we examine the effects of noise and the subthreshold dynamic properties of the RIF model on the occurrence of coherence resonance (CR). Synchronized burst firing occurs in a network of such model neurons with excitatory pulse-coupling. The coherence level of the network oscillations shows a stochastic resonance-like dependence on the noise level. Stochastic analysis of the equations shows that the slow recovery from the spike-induced inhibition is crucial in determining the frequencies of the CR and the subthreshold resonance in the original HR model. In this particular type of CR, the oscillation frequency strongly depends on the intrinsic time scales but changes little with the noise intensity. We give analytical quantities to describe this CR mechanism and illustrate its influence on the emerging network oscillations. We discuss the profound physiological roles this kind of CR may have in information processing in neurons possessing a subthreshold resonant frequency and in generating synchronized network oscillations with a frequency that is determined by intrinsic properties of the neurons.
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Affiliation(s)
- Stefan Reinker
- Department of Mathematics and Physics, FDM, University of Freiburg, Hermann-Herder-Str. 3, 79104 Freiburg, Germany.
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32
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Sacerdote L, Villa AEP, Zucca C. On the Classification of Experimental Data Modeled Via a Stochastic Leaky Integrate and Fire Model Through Boundary Values. Bull Math Biol 2006; 68:1257-74. [PMID: 17149816 DOI: 10.1007/s11538-006-9107-7] [Citation(s) in RCA: 10] [Impact Index Per Article: 0.6] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/28/2022]
Abstract
We present a computational algorithm aimed to classify single unit spike trains on the basis of observed interspikes intervals (ISI). The neuronal activity is modeled with a stochastic leaky integrate and fire model and the inverse first passage time method is extended to the Ornstein-Uhlenbeck (OU) process. Differences between spike trains are detected in terms of the boundary shape. The proposed classification method is applied to the analysis of multiple single units recorded simultaneously in the thalamus and in the cerebral cortex of unanesthetized rats during spontaneous activity. We show the existence of at least three different firing patterns that could not be classified using the usual statistical indices.
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Affiliation(s)
- L Sacerdote
- Department of Mathematics, University of Torino, Via Carlo Alberto 10, 10123 Torino, Italy.
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33
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Sakumura Y, Ishii S. Stochastic resonance with differential code in feedforward network with intra-layer random connections. Neural Netw 2005; 19:469-76. [PMID: 16150572 DOI: 10.1016/j.neunet.2005.05.002] [Citation(s) in RCA: 2] [Impact Index Per Article: 0.1] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 12/13/2004] [Accepted: 05/31/2005] [Indexed: 10/25/2022]
Abstract
We examined stochastic resonance with a differential coding scheme using a multilayer feedforward neural network which is composed of intra-layer connections. We show that the network, with random synaptic connections in each layer, encodes an input signal into a spike coherence that represents temporal differences among the inputs. We also demonstrate that both internal and external noise enhance the detection of weak signals. Finally, we discuss how the feedforward network with intra-layer random connections is similar to a membrane in its sensitivity to and amplification of a change in stimulus and suggest that the intensity of internal noise may be tuned in a real brain.
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Affiliation(s)
- Yuichi Sakumura
- Graduate School of Information Science, Nara Institute of Science and Technology, 8916-5 Takayama, Ikoma, Nara 630-0192, Japan
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34
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Abstract
In this paper, two stochastic versions of the LIF neural model are considered: one with the noise signal applied to the firing threshold, the other having it added to the input current. Then, adopting a discontinuous stepwise noise whose innovations are uncorrelated and gaussian distributed, the behaviours of the two models pertaining to the stochastic resonance (SR) are analysed and compared. Furthermore, it is shown that introducing a suitable time correlation into the noise signal brings us from the first model to the second one.
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Affiliation(s)
- Michele Barbi
- Istituto di Biofisica del, CNR Via G. Moruzzi 1, 56124 Pisa, Italy.
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35
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Greenwood PE, Lánský P. Optimum signal in a simple neuronal model with signal-dependent noise. BIOLOGICAL CYBERNETICS 2005; 92:199-205. [PMID: 15750866 DOI: 10.1007/s00422-005-0545-3] [Citation(s) in RCA: 11] [Impact Index Per Article: 0.6] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Subscribe] [Scholar Register] [Received: 06/24/2004] [Accepted: 01/10/2005] [Indexed: 05/24/2023]
Abstract
How does the information about a signal in neural threshold crossings depend on the noise acting upon it? Two models are explored, a binary McCulloch and Pitts (threshold exceedance) model and a model of waiting time to exceedance--a discrete-time version of interspike intervals. If noise grows linearly with the signal, we find the best identification of the signal in terms of the Fisher information is for signals that do not reach the threshold in the absence of noise. Identification attains the same precision under weak and strong signals, but the coding range decreases at both extremes of signal level. We compare the results obtained for Fisher information with those using related first and second moment measures. The maximum obtainable information is plotted as a function of the ratio of noise to signal.
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Affiliation(s)
- Priscilla E Greenwood
- Department of Mathematics and Statistics, Arizona State University, Tempe, AZ 85287, USA
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36
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Wenning G, Hoch T, Obermayer K. Detection of pulses in a colored noise setting. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2005; 71:021902. [PMID: 15783347 DOI: 10.1103/physreve.71.021902] [Citation(s) in RCA: 7] [Impact Index Per Article: 0.4] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Subscribe] [Scholar Register] [Received: 06/23/2004] [Indexed: 05/24/2023]
Abstract
Cortical neurons are exposed to a considerable amount of synaptic background activity, which increases the neurons' conductance and which leads to a fluctuating membrane potential. Here we investigate how the presence and the properties of this background noise influence the ability of a neuron to detect transient inputs, a task that is important for coincidence detection as well as for the detection of synchronous spiking events in a neural system. Using a leaky integrate-and-fire neuron as well as a biologically more realistic Hodgkin-Huxley type point neuron we find that noise enhances the detection of subthreshold input pulses and that the phenomenon of stochastic resonance occurs. When the noise is colored, pulse detection becomes more robust, because the number of false positive events decreases with increasing temporal correlation while the number of correctly detected events is almost unaffected. Therefore, the optimal variance of the noise also changes with the degree of temporal correlations of the background activity. For the integrate-and-fire model these effects can be described using an ansatz by Brunel and Sergi [J. Theor. Biol. 195, 87 (1998)]. Numerical simulations show that the leaky integrate-and-fire model and the Hodgkin-Huxley type point neuron behave qualitatively similarly.
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Affiliation(s)
- Gregor Wenning
- Department of Electrical Engineering and Computer Science, Technical University of Berlin, Franklinstrasse 28/29, 10587 Berlin, Germany
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37
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Giraudo MT, Sacerdote L. Effect of periodic stimulus on a neuronal diffusion model with signal-dependent noise. Biosystems 2005; 79:73-81. [PMID: 15649591 DOI: 10.1016/j.biosystems.2004.09.021] [Citation(s) in RCA: 5] [Impact Index Per Article: 0.3] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/16/2022]
Abstract
To relate the noise intensity with a periodically modulated input signal in a single neuron stochastic model we introduce a diffusion model with both time modulated drift and diffusion coefficient. Such a model is the continuous version of a Stein model with time oscillating frequencies for the Poisson processes describing the inputs impinging on the neuron. We focus here on some aspects of the resonance phenomenon for such a model. We compare the corresponding interspike interval distribution with the analogous distribution for a model sharing the same parameter values, but with constant noise intensity. Examples with two different levels for this noise intensity are discussed. The enhancement of the height of the peaks in the interspike interval distribution appearing at the modulation period, the improvement of the phase locking behavior and an enlargement of the noise ranges where a resonance like behavior arises are the main features observed in the considered cases.
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Affiliation(s)
- Maria Teresa Giraudo
- Department of Mathematics, University of Torino, V.C. Alberto 10, 10123 Torino, Italy.
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38
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Ditlevsen S, Lansky P. Estimation of the input parameters in the Ornstein-Uhlenbeck neuronal model. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2005; 71:011907. [PMID: 15697630 DOI: 10.1103/physreve.71.011907] [Citation(s) in RCA: 32] [Impact Index Per Article: 1.7] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Subscribe] [Scholar Register] [Received: 08/17/2004] [Indexed: 05/24/2023]
Abstract
The stochastic Ornstein-Uhlenbeck neuronal model is studied, and estimators of the model input parameters, depending on the firing regime of the process, are derived. Closed expressions for the Laplace transforms of the first two moments of the normalized first-passage time through a constant boundary in the suprathreshold regime are derived, which is used to define moment estimators. In the subthreshold regime, the exponentiality of the first-passage time is utilized to characterize the input parameters. In the threshold regime and for the Wiener process approximation, analytic expressions for the first-passage-time density are used to derive the maximum-likelihood estimators of the parameters. The methods are illustrated on simulated data under different conditions, including misspecification of the intrinsic parameters of the model. Finally, known approximations of the first-passage-time moments are improved.
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Affiliation(s)
- Susanne Ditlevsen
- Department of Biostatistics, Panum Institute, University of Copenhagen, Blegdamsvej 3, 2200 N, Denmark.
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39
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Mitaim S, Kosko B. Adaptive Stochastic Resonance in Noisy Neurons Based on Mutual Information. ACTA ACUST UNITED AC 2004; 15:1526-40. [PMID: 15565779 DOI: 10.1109/tnn.2004.826218] [Citation(s) in RCA: 88] [Impact Index Per Article: 4.4] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/10/2022]
Abstract
Noise can improve how memoryless neurons process signals and maximize their throughput information. Such favorable use of noise is the so-called "stochastic resonance" or SR effect at the level of threshold neurons and continuous neurons. This paper presents theoretical and simulation evidence that 1) lone noisy threshold and continuous neurons exhibit the SR effect in terms of the mutual information between random input and output sequences, 2) a new statistically robust learning law can find this entropy-optimal noise level, and 3) the adaptive SR effect is robust against highly impulsive noise with infinite variance. Histograms estimate the relevant probability density functions at each learning iteration. A theorem shows that almost all noise probability density functions produce some SR effect in threshold neurons even if the noise is impulsive and has infinite variance. The optimal noise level in threshold neurons also behaves nonlinearly as the input signal amplitude increases. Simulations further show that the SR effect persists for several sigmoidal neurons and for Gaussian radial-basis-function neurons.
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Affiliation(s)
- Sanya Mitaim
- Department of Electrical Engineering, Faculty of Engineering, Thammasat University, Rangsit Campus, Klong Luang, Pathumthani 12120, Thailand.
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40
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Bloodworth DM, Nguyen BN, Garver W, Moss F, Pedroza C, Tran T, Chiou-Tan FY. Comparison of Stochastic vs. Conventional Transcutaneous Electrical Stimulation for Pain Modulation in Patients with Electromyographically Documented Radiculopathy. Am J Phys Med Rehabil 2004; 83:584-91. [PMID: 15277959 DOI: 10.1097/01.phm.0000133439.28817.51] [Citation(s) in RCA: 24] [Impact Index Per Article: 1.2] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/27/2022]
Abstract
OBJECTIVE To determine if a transcutaneous electrical stimulation (TENS) unit modified to deliver electrical impulses at random (R) or stochastic frequency, called TENS-R, provided better pain relief than conventional TENS. DESIGN A prospective, randomized, double-blinded, placebo-controlled study at an urban teaching hospital. A total of 13 adult subjects with radiculopathy on electromyogram and chronic radicular pain rated pain before and after walking 100 feet with proximal (axial) placement of TENS leads with randomized settings on conventional TENS, placebo, or TENS-R and, subsequently, with distal (limb) placement of TENS leads with randomized settings, all on the same day. The pain measures used were the McGill Pain Questionnaire, parts 1 and 2, and the Visual Analog Scale. The functional measure was speed of walking. RESULTS Four men and seven women completed the study pain scores, measured by McGill Pain Questionnaire part 2, significantly improved when the patient used TENS-R vs. conventional TENS (P = 0.006, analysis of variance). Placement of TENS electrodes on the back significantly decreased pain compared with lead placement on the legs for McGill Pain Questionnaire part 1 (P = 0.007), McGill Pain Questionnaire part 2 (P = 0.042), and the Visual Analog Scale (P = 0.026) measures. CONCLUSIONS Qualitative pain scores significantly improved when the patient used TENS-R vs. conventional TENS. Lead placement of any TENS modality over the back vs. over the leg improved all pain scores.
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Affiliation(s)
- Donna M Bloodworth
- Department of Physical Medicine and Rehabilitation, Baylor College of Medicine, 3601 North MacGregor Way, Room 240B, Houston, TX 77004, USA
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Schindler M, Talkner P, Hänggi P. Firing time statistics for driven neuron models: analytic expressions versus numerics. PHYSICAL REVIEW LETTERS 2004; 93:048102. [PMID: 15323796 DOI: 10.1103/physrevlett.93.048102] [Citation(s) in RCA: 11] [Impact Index Per Article: 0.6] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Subscribe] [Scholar Register] [Received: 01/09/2004] [Indexed: 05/24/2023]
Abstract
Analytical expressions are put forward to investigate the forced spiking activity of abstract neuron models such as the driven leaky integrate-and-fire model. The method is valid in a wide parameter regime beyond the restraining limits of weak driving (linear response) and/or weak noise. The novel approximation is based on a discrete state Markovian modeling of the full long-time dynamics with time-dependent rates. The scheme yields excellent agreement with numerical Langevin and Fokker-Planck simulations of the full nonstationary dynamics, not only for the first-passage time statistics, but also for the important interspike interval (residence time) distribution.
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Affiliation(s)
- Michael Schindler
- Institut für Physik, Universität Augsburg, Universitätsstrasse 1, D-86135 Augsburg, Germany
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Barbi M, Chillemi S, Garbo AD, Reale L. Stochastic resonance in a sinusoidally forced LIF model with noisy threshold. Biosystems 2003; 71:23-8. [PMID: 14568203 DOI: 10.1016/s0303-2647(03)00106-0] [Citation(s) in RCA: 5] [Impact Index Per Article: 0.2] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/27/2022]
Abstract
In this report, the LIF neural model driven by underthreshold sinusoidal signals but with a gaussian-distributed noise on the threshold, is approximated by suitably defining an instantaneous firing (or escape) rate, which depends only on the momentary value of the voltage variable. This allows us to obtain, by analytically solving the relevant equations, the main statistical functions describing the "firing activity"; namely, the probability density function of firing phases and that of interspike intervals. From these functions two quantities can be derived, whose dependence on the noise intensity allows the Stochastic Resonance (SR) to be demonstrated. Besides the "regular" SR, the analysed system was found to produce, either for low frequencies and large amplitudes of modulation or for high modulation frequencies, resonance curves displaying two peaks. This bimodal feature of the resonance curves is accounted for on the basis of phase locked firing patterns.
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Affiliation(s)
- Michele Barbi
- Istituto di Biofisica del CNR, Via G. Moruzzi 1, Pisa 56124, Italy.
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Jianfeng Feng, Yunlian Sun, Buxton H, Gang Wei. Training integrate-and-fire neurons with the informax principle II. ACTA ACUST UNITED AC 2003; 14:326-36. [DOI: 10.1109/tnn.2003.809419] [Citation(s) in RCA: 9] [Impact Index Per Article: 0.4] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/08/2022]
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Abstract
A recent computational study of gain control via shunting inhibition has shown that the slope of the frequency-versus-input (f-I) characteristic of a neuron can be decreased by increasing the noise associated with the inhibitory input (Neural Comput. 13, 227-248). This novel noise-induced divisive gain control relies on the concommittant increase of the noise variance with the mean of the total inhibitory conductance. Here we investigate this effect using different neuronal models. The effect is shown to occur in the standard leaky integrate-and-fire (LIF) model with additive Gaussian white noise, and in the LIF with multiplicative noise acting on the inhibitory conductance. The noisy scaling of input currents is also shown to occur in the one-dimensional theta-neuron model, which has firing dynamics, as well as a large scale compartmental model of a pyramidal cell in the electrosensory lateral line lobe of a weakly electric fish. In this latter case, both the inhibition and the excitatory input have Poisson statistics; noise-induced divisive inhibition is thus seen in f-I curves for which the noise increases along with the input I. We discuss how the variation of the noise intensity along with inputs is constrained by the physiological context and the class of model used, and further provide a comparison of the divisive effect across models.
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Affiliation(s)
- André Longtin
- Department of Physics, University of Ottawa, 150 Louis Pasteur, Ottawa, Ont., Canada K1N 6N5.
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Sacerdote L, Lánský P. Interspike interval statistics in the Ornstein-Uhlenbeck neuronal model with signal-dependent noise. Biosystems 2002; 67:213-9. [PMID: 12459301 DOI: 10.1016/s0303-2647(02)00079-5] [Citation(s) in RCA: 4] [Impact Index Per Article: 0.2] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/23/2022]
Abstract
The stochastic leaky integrate-and-fire (LIF) continuous model is studied under the condition that the amplitude of noise is a function of the input signal. The coefficient of variation (CV) of interspike intervals (ISIs) is investigated for different types of dependencies between the noise and the signal. Finally, we present the CV and the ISI density resulting from the special choice of parameters of the input that gave rise to a contra-intuitive behavior of the transfer function in Lánský and Sacerdote [Phys. Lett. A 285 (2001) 132].
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Affiliation(s)
- Laura Sacerdote
- Dipartimento di Matematica, Università di Torino, Via Carlo Alberto 10, 10123 Turin, Italy.
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Lindner B, Schimansky-Geier L, Longtin A. Maximizing spike train coherence or incoherence in the leaky integrate-and-fire model. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2002; 66:031916. [PMID: 12366161 DOI: 10.1103/physreve.66.031916] [Citation(s) in RCA: 90] [Impact Index Per Article: 4.1] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 01/25/2002] [Revised: 04/08/2002] [Indexed: 05/23/2023]
Abstract
We study noise-induced resonance effects in the leaky integrate-and-fire neuron model with absolute refractory period, driven by a Gaussian white noise. It is demonstrated that a finite noise level may either maximize or minimize the regularity of the spike train. We also partition the parameter space into regimes where either or both of these effects occur. It is shown that the coherence minimization at moderate noise results in a flat spectral response with respect to periodic stimulation in contrast to sharp resonances that are observed for both small and large noise intensities.
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Affiliation(s)
- Benjamin Lindner
- Institute of Physics, Humboldt-University at Berlin, Invalidenstrasse 110, D-10115 Berlin, Germany
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Choi MH, Fox RF. Evolution of escape processes with a time-varying load. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2002; 66:031103. [PMID: 12366095 DOI: 10.1103/physreve.66.031103] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 03/29/2002] [Indexed: 05/23/2023]
Abstract
We study an escape process of a noisy particle with a time varying load. We present an effective nonperturbative method which works even when the time varying load amplitude is comparable to other parameters. It is based on the idea that for every instant of time, we know the quasiadiabatic eigenspectrum and the quasiadiabatic eigenfunctions of the instantaneous system. We show that when two time-varying quasiadiabatic eigenvalues in the spectrum get close to each other, the amplitudes of the quasiadiabatic eigenfuntions show an abrupt change; therefore, the escape rate is highly affected.
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Affiliation(s)
- Mee H Choi
- School of Physics, Georgia Institute of Technology, Atlanta, Georgia 30332, USA
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Hasegawa H. Stochastic resonance of ensemble neurons for transient spike trains: wavelet analysis. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2002; 66:021902. [PMID: 12241209 DOI: 10.1103/physreve.66.021902] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.0] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Subscribe] [Scholar Register] [Received: 11/28/2001] [Indexed: 05/23/2023]
Abstract
By using the wavelet transformation (WT), I have analyzed the response of an ensemble of N (=1, 10, 100, and 500) Hodgkin-Huxley neurons to transient M-pulse spike trains (M=1 to 3) with independent Gaussian noises. The cross correlation between the input and output signals is expressed in terms of the WT expansion coefficients. The signal-to-noise ratio (SNR) is evaluated by using the denoising method within the WT, by which the noise contribution is extracted from the output signals. Although the response of a single (N=1) neuron to subthreshold transient signals with noises is quite unreliable, the transmission fidelity assessed by the cross correlation and SNR is shown to be much improved by increasing the value of N: a population of neurons plays an indispensable role in the stochastic resonance (SR) for transient spike inputs. It is also shown that in a large-scale ensemble, the transmission fidelity for suprathreshold transient spikes is not significantly degraded by a weak noise which is responsible to SR for subthreshold inputs.
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Affiliation(s)
- Hideo Hasegawa
- Department of Physics, Tokyo Gakugei University, Koganei, Tokyo 184-8501, Japan.
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Abstract
The reliability of firing of excitable-oscillating systems is studied through the response of the active rotator, a neuronal model evolving on the unit circle, to white gaussian noise. A stochastic return map is introduced that captures the behavior of the model. This map has two fixed points: one stable and the other unstable. Iterates of all initial conditions except the unstable point tend to the stable fixed point for almost all input realizations. This means that to a given input realization, there corresponds a unique asymptotic response. In this way, repetitive stimulation with the same segment of noise realization evokes, after possibly a transient time, the same response in the active rotator. In other words, this model responds reliably to such inputs. It is argued that this results from the nonuniform motion of the active rotator around the unit circle and that similar results hold for other neuronal models whose dynamics can be approximated by phase dynamics similar to the active rotator.
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Affiliation(s)
- K Pakdaman
- Inserm U444, Faculté de Médecine Saint-Antoine, 75571 Paris Cedex 12, France.
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Abstract
The timing information contained in the response of a neuron to noisy periodic synaptic input is analyzed for the leaky integrate-and-fire neural model. We address the question of the relationship between the timing of the synaptic inputs and the output spikes. This requires an analysis of the interspike interval distribution of the output spikes, which is obtained in the gaussian approximation. The conditional output spike density in response to noisy periodic input is evaluated as a function of the initial phase of the inputs. This enables the phase transition matrix to be calculated, which relates the phase at which the output spike is generated to the initial phase of the inputs. The interspike interval histogram and the period histogram for the neural response to ongoing periodic input are then evaluated by using the leading eigenvector of this phase transition matrix. The synchronization index of the output spikes is found to increase sharply as the inputs become synchronized. This enhancement of synchronization is most pronounced for large numbers of inputs and lower frequencies of modulation and also for rates of input near the critical input rate. However, the mutual information between the input phase of the stimulus and the timing of output spikes is found to decrease at low input rates as the number of inputs increases. The results show close agreement with those obtained from numerical simulations for large numbers of inputs.
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Affiliation(s)
- A N Burkitt
- Bionic Ear Institute, East Melbourne, Victoria 3002, Australia.
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