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Ramlow L, Falcke M, Lindner B. An integrate-and-fire approach to Ca 2+ signaling. Part II: Cumulative refractoriness. Biophys J 2023; 122:4710-4729. [PMID: 37981761 PMCID: PMC10754692 DOI: 10.1016/j.bpj.2023.11.015] [Citation(s) in RCA: 1] [Impact Index Per Article: 1.0] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 07/25/2023] [Revised: 10/20/2023] [Accepted: 11/15/2023] [Indexed: 11/21/2023] Open
Abstract
Inositol 1,4,5-trisphosphate-induced Ca2+ signaling is a second messenger system used by almost all eukaryotic cells. The agonist concentration stimulating Ca2+ signals is encoded in the frequency of a Ca2+ concentration spike sequence. When a cell is stimulated, the interspike intervals (ISIs) often show a distinct transient during which they gradually increase, a system property we refer to as cumulative refractoriness. We extend a previously published stochastic model to include the Ca2+ concentration in the intracellular Ca2+ store as a slow adaptation variable. This model can reproduce both stationary and transient statistics of experimentally observed ISI sequences. We derive approximate expressions for the mean and coefficient of variation of the stationary ISIs. We also consider the response to the onset of a constant stimulus and estimate the length of the transient and the strength of the adaptation of the ISI. We show that the adaptation sets the coefficient of variation in agreement with current ideas derived from experiments. Moreover, we explain why, despite a pronounced transient behavior, ISI correlations can be weak, as often observed in experiments. Finally, we fit our model to reproduce the transient statistics of experimentally observed ISI sequences in stimulated HEK cells. The fitted model is able to qualitatively reproduce the relationship between the stationary interval correlations and the number of transient intervals, as well as the strength of the ISI adaptation. We also find positive correlations in the experimental sequence that cannot be explained by our model.
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Affiliation(s)
- Lukas Ramlow
- Bernstein Center for Computational Neuroscience Berlin, Berlin, Germany; Department of Physics, Humboldt University Berlin, Berlin, Germany; Max Delbrück Center for Molecular Medicine, Berlin, Germany
| | - Martin Falcke
- Department of Physics, Humboldt University Berlin, Berlin, Germany; Max Delbrück Center for Molecular Medicine, Berlin, Germany.
| | - Benjamin Lindner
- Bernstein Center for Computational Neuroscience Berlin, Berlin, Germany; Department of Physics, Humboldt University Berlin, Berlin, Germany
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2
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Sidhu RS, Johnson EC, Jones DL, Ratnam R. A dynamic spike threshold with correlated noise predicts observed patterns of negative interval correlations in neuronal spike trains. BIOLOGICAL CYBERNETICS 2022; 116:611-633. [PMID: 36244004 PMCID: PMC9691502 DOI: 10.1007/s00422-022-00946-5] [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: 12/10/2021] [Accepted: 09/26/2022] [Indexed: 06/16/2023]
Abstract
Negative correlations in the sequential evolution of interspike intervals (ISIs) are a signature of memory in neuronal spike-trains. They provide coding benefits including firing-rate stabilization, improved detectability of weak sensory signals, and enhanced transmission of information by improving signal-to-noise ratio. Primary electrosensory afferent spike-trains in weakly electric fish fall into two categories based on the pattern of ISI correlations: non-bursting units have negative correlations which remain negative but decay to zero with increasing lags (Type I ISI correlations), and bursting units have oscillatory (alternating sign) correlation which damp to zero with increasing lags (Type II ISI correlations). Here, we predict and match observed ISI correlations in these afferents using a stochastic dynamic threshold model. We determine the ISI correlation function as a function of an arbitrary discrete noise correlation function [Formula: see text], where k is a multiple of the mean ISI. The function permits forward and inverse calculations of the correlation function. Both types of correlation functions can be generated by adding colored noise to the spike threshold with Type I correlations generated with slow noise and Type II correlations generated with fast noise. A first-order autoregressive (AR) process with a single parameter is sufficient to predict and accurately match both types of afferent ISI correlation functions, with the type being determined by the sign of the AR parameter. The predicted and experimentally observed correlations are in geometric progression. The theory predicts that the limiting sum of ISI correlations is [Formula: see text] yielding a perfect DC-block in the power spectrum of the spike train. Observed ISI correlations from afferents have a limiting sum that is slightly larger at [Formula: see text] ([Formula: see text]). We conclude that the underlying process for generating ISIs may be a simple combination of low-order AR and moving average processes and discuss the results from the perspective of optimal coding.
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Affiliation(s)
- Robin S Sidhu
- Department of Electrical and Computer Engineering, University of Illinois at Urbana-Champaign, Urbana, IL, USA
| | - Erik C Johnson
- The Johns Hopkins University Applied Physics Laboratory, Laurel, MD, USA
| | - Douglas L Jones
- Department of Electrical and Computer Engineering, University of Illinois at Urbana-Champaign, Urbana, IL, USA
| | - Rama Ratnam
- Division of Biological and Life Sciences, School of Arts and Sciences, Ahmedabad University, Ahmedabad, Gujarat, India.
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3
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Ramlow L, Lindner B. Interspike interval correlations in neuron models with adaptation and correlated noise. PLoS Comput Biol 2021; 17:e1009261. [PMID: 34449771 PMCID: PMC8428727 DOI: 10.1371/journal.pcbi.1009261] [Citation(s) in RCA: 2] [Impact Index Per Article: 0.7] [Reference Citation Analysis] [Abstract] [MESH Headings] [Grants] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 05/28/2020] [Revised: 09/09/2021] [Accepted: 07/08/2021] [Indexed: 11/19/2022] Open
Abstract
The generation of neural action potentials (spikes) is random but nevertheless may result in a rich statistical structure of the spike sequence. In particular, contrary to the popular renewal assumption of theoreticians, the intervals between adjacent spikes are often correlated. Experimentally, different patterns of interspike-interval correlations have been observed and computational studies have identified spike-frequency adaptation and correlated noise as the two main mechanisms that can lead to such correlations. Analytical studies have focused on the single cases of either correlated (colored) noise or adaptation currents in combination with uncorrelated (white) noise. For low-pass filtered noise or adaptation, the serial correlation coefficient can be approximated as a single geometric sequence of the lag between the intervals, providing an explanation for some of the experimentally observed patterns. Here we address the problem of interval correlations for a widely used class of models, multidimensional integrate-and-fire neurons subject to a combination of colored and white noise sources and a spike-triggered adaptation current. Assuming weak noise, we derive a simple formula for the serial correlation coefficient, a sum of two geometric sequences, which accounts for a large class of correlation patterns. The theory is confirmed by means of numerical simulations in a number of special cases including the leaky, quadratic, and generalized integrate-and-fire models with colored noise and spike-frequency adaptation. Furthermore we study the case in which the adaptation current and the colored noise share the same time scale, corresponding to a slow stochastic population of adaptation channels; we demonstrate that our theory can account for a nonmonotonic dependence of the correlation coefficient on the channel's time scale. Another application of the theory is a neuron driven by network-noise-like fluctuations (green noise). We also discuss the range of validity of our weak-noise theory and show that by changing the relative strength of white and colored noise sources, we can change the sign of the correlation coefficient. Finally, we apply our theory to a conductance-based model which demonstrates its broad applicability.
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Affiliation(s)
- Lukas Ramlow
- Bernstein Center for Computational Neuroscience Berlin, Berlin, Germany
- Physics Department, Humboldt University zu Berlin, Berlin, Germany
| | - Benjamin Lindner
- Bernstein Center for Computational Neuroscience Berlin, Berlin, Germany
- Physics Department, Humboldt University zu Berlin, Berlin, Germany
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4
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Yu Z, Guo S, Deng F, Yan Q, Huang K, Liu JK, Chen F. Emergent Inference of Hidden Markov Models in Spiking Neural Networks Through Winner-Take-All. IEEE TRANSACTIONS ON CYBERNETICS 2020; 50:1347-1354. [PMID: 30295641 DOI: 10.1109/tcyb.2018.2871144] [Citation(s) in RCA: 8] [Impact Index Per Article: 2.0] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 06/08/2023]
Abstract
Hidden Markov models (HMMs) underpin the solution to many problems in computational neuroscience. However, it is still unclear how to implement inference of HMMs with a network of neurons in the brain. The existing methods suffer from the problem of being nonspiking and inaccurate. Here, we build a precise equivalence between the inference equation of HMMs with time-invariant hidden variables and the dynamics of spiking winner-take-all (WTA) neural networks. We show that the membrane potential of each spiking neuron in the WTA circuit encodes the logarithm of the posterior probability of the hidden variable in each state, and the firing rate of each neuron is proportional to the posterior probability of the HMMs. We prove that the time course of the neural firing rate can implement posterior inference of HMMs. Theoretical analysis and experimental results show that the proposed WTA circuit can get accurate inference results of HMMs.
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5
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Braun W, Longtin A. Interspike interval correlations in networks of inhibitory integrate-and-fire neurons. Phys Rev E 2019; 99:032402. [PMID: 30999498 DOI: 10.1103/physreve.99.032402] [Citation(s) in RCA: 2] [Impact Index Per Article: 0.4] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 12/20/2018] [Indexed: 11/07/2022]
Abstract
We study temporal correlations of interspike intervals, quantified by the network-averaged serial correlation coefficient (SCC), in networks of both current- and conductance-based purely inhibitory integrate-and-fire neurons. Numerical simulations reveal transitions to negative SCCs at intermediate values of bias current drive and network size. As bias drive and network size are increased past these values, the SCC returns to zero. The SCC is maximally negative at an intermediate value of the network oscillation strength. The dependence of the SCC on two canonical schemes for synaptic connectivity is studied, and it is shown that the results occur robustly in both schemes. For conductance-based synapses, the SCC becomes negative at the onset of both a fast and slow coherent network oscillation. We then show by means of offline simulations using prerecorded network activity that a neuron's SCC is highly sensitive to its number of presynaptic inputs. Finally, we devise a noise-reduced diffusion approximation for current-based networks that accounts for the observed temporal correlation transitions.
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Affiliation(s)
- Wilhelm Braun
- Neural Network Dynamics and Computation, Institut für Genetik, Universität Bonn, Kirschallee 1, 53115 Bonn, Germany.,Department of Physics and Centre for Neural Dynamics, University of Ottawa, 598 King Edward, Ottawa K1N 6N5, Canada
| | - André Longtin
- Department of Physics and Centre for Neural Dynamics, University of Ottawa, 598 King Edward, Ottawa K1N 6N5, Canada
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6
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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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7
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Mankin R, Rekker A. Response to a periodic stimulus in a perfect integrate-and-fire neuron model driven by colored noise. Phys Rev E 2016; 94:062103. [PMID: 28085436 DOI: 10.1103/physreve.94.062103] [Citation(s) in RCA: 4] [Impact Index Per Article: 0.5] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 07/14/2016] [Indexed: 06/06/2023]
Abstract
The output interspike interval statistics of a stochastic perfect integrate-and-fire neuron model driven by an additive exogenous periodic stimulus is considered. The effect of temporally correlated random activity of synaptic inputs is modeled by an additive symmetric dichotomous noise. Using a first-passage-time formulation, exact expressions for the output interspike interval density and for the serial correlation coefficient are derived in the nonsteady regime, and their dependence on input parameters (e.g., the noise correlation time and amplitude as well as the frequency of an input current) is analyzed. It is shown that an interplay of a periodic forcing and colored noise can cause a variety of nonequilibrium cooperation effects, such as sign reversals of the interspike interval correlations versus noise-switching rate as well as versus the frequency of periodic forcing, a power-law-like decay of oscillations of the serial correlation coefficients in the long-lag limit, amplification of the output signal modulation in the instantaneous firing rate of the neural response, etc. The features of spike statistics in the limits of slow and fast noises are also discussed.
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Affiliation(s)
- Romi Mankin
- School of Natural Sciences and Health, Tallinn University, 29 Narva Road, 10120 Tallinn, Estonia
| | - Astrid Rekker
- School of Natural Sciences and Health, Tallinn University, 29 Narva Road, 10120 Tallinn, Estonia
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8
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Ryabov A, Zemánek P, Filip R. Thermally induced passage and current of particles in a highly unstable optical potential. Phys Rev E 2016; 94:042108. [PMID: 27841491 DOI: 10.1103/physreve.94.042108] [Citation(s) in RCA: 2] [Impact Index Per Article: 0.3] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 05/28/2016] [Indexed: 06/06/2023]
Abstract
We discuss the statistics of first-passage times of a Brownian particle moving in a highly unstable nonlinear potential proportional to an odd power of position. We observe temperature-induced shortening of the mean first-passage time and its dependence on the power of nonlinearity. We propose a passage-time fraction as both a simple and experimentally detectable witness of the nonlinearity. It is advantageously independent of all other parameters of the experiment and observable for a small number of trajectories. To better characterize the stochastic passage in the unstable potential, we introduce an analogy of the signal-to-noise ratio for the statistical distribution of the first-passage times. Interestingly, the upper bound for the signal-to-noise ratio is temperature independent in the unstable potential. Finally, we describe the nonequilibrium steady state of the particle cyclically passing through unstable odd nonlinearity. The maximum of the steady-state probability distribution shifts against the directions of the current and this counterintuitive effect increases with temperature. All these thermally induced effects are very promising targets for experimental tests of highly nonlinear stochastic dynamics of particles placed into optical potential landscapes of shaped optical tweezers.
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Affiliation(s)
- Artem Ryabov
- Charles University in Prague, Faculty of Mathematics and Physics, Department of Macromolecular Physics, V Holešovičkách 2, 180 00 Praha 8, Czech Republic
| | - Pavel Zemánek
- Institute of Scientific Instruments of CAS, Czech Academy of Sciences, Královopolská 147, 612 64 Brno, Czech Republic
| | - Radim Filip
- Department of Optics, Palacký University, 17 Listopadu 1192/12, 771 46 Olomouc, Czech Republic
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9
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Heil P, Peterson AJ. Spike timing in auditory-nerve fibers during spontaneous activity and phase locking. Synapse 2016; 71:5-36. [DOI: 10.1002/syn.21925] [Citation(s) in RCA: 28] [Impact Index Per Article: 3.5] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 12/01/2015] [Revised: 07/20/2016] [Accepted: 07/24/2016] [Indexed: 12/22/2022]
Affiliation(s)
- Peter Heil
- Department of Systems Physiology of Learning; Leibniz Institute for Neurobiology; Magdeburg 39118 Germany
- Center for Behavioral Brain Sciences; Magdeburg Germany
| | - Adam J. Peterson
- Department of Systems Physiology of Learning; Leibniz Institute for Neurobiology; Magdeburg 39118 Germany
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10
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Schwalger T, Lindner B. Analytical approach to an integrate-and-fire model with spike-triggered adaptation. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2015; 92:062703. [PMID: 26764723 DOI: 10.1103/physreve.92.062703] [Citation(s) in RCA: 4] [Impact Index Per Article: 0.4] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 05/31/2015] [Indexed: 06/05/2023]
Abstract
The calculation of the steady-state probability density for multidimensional stochastic systems that do not obey detailed balance is a difficult problem. Here we present the analytical derivation of the stationary joint and various marginal probability densities for a stochastic neuron model with adaptation current. Our approach assumes weak noise but is valid for arbitrary adaptation strength and time scale. The theory predicts several effects of adaptation on the statistics of the membrane potential of a tonically firing neuron: (i) a membrane potential distribution with a convex shape, (ii) a strongly increased probability of hyperpolarized membrane potentials induced by strong and fast adaptation, and (iii) a maximized variability associated with the adaptation current at a finite adaptation time scale.
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Affiliation(s)
- Tilo Schwalger
- Brain Mind Institute, École Polytechnique Féderale de Lausanne (EPFL) Station 15, CH-1015 Lausanne, Switzerland
- Bernstein Center for Computational Neuroscience, Haus 2, Philippstraße 13, 10115 Berlin, Germany
| | - Benjamin Lindner
- Bernstein Center for Computational Neuroscience, Haus 2, Philippstraße 13, 10115 Berlin, Germany
- Department of Physics, Humboldt Universität zu Berlin, Newtonstraße 15, 12489 Berlin, Germany
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11
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Ryabov A, Berestneva E, Holubec V. Brownian motion in time-dependent logarithmic potential: Exact results for dynamics and first-passage properties. J Chem Phys 2015; 143:114117. [PMID: 26395697 DOI: 10.1063/1.4931474] [Citation(s) in RCA: 14] [Impact Index Per Article: 1.6] [Reference Citation Analysis] [MESH Headings] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/14/2022] Open
Affiliation(s)
- Artem Ryabov
- Charles University in Prague, Faculty of Mathematics and Physics, Department of Macromolecular Physics, V Holešovičkách 2, 180 00 Praha 8, Czech Republic
| | - Ekaterina Berestneva
- Charles University in Prague, Faculty of Mathematics and Physics, Department of Macromolecular Physics, V Holešovičkách 2, 180 00 Praha 8, Czech Republic
| | - Viktor Holubec
- Charles University in Prague, Faculty of Mathematics and Physics, Department of Macromolecular Physics, V Holešovičkách 2, 180 00 Praha 8, Czech Republic
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12
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Urdapilleta E, Samengo I. Effects of spike-triggered negative feedback on receptive-field properties. J Comput Neurosci 2015; 38:405-25. [PMID: 25601482 DOI: 10.1007/s10827-014-0546-0] [Citation(s) in RCA: 3] [Impact Index Per Article: 0.3] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 12/08/2014] [Accepted: 12/30/2014] [Indexed: 11/29/2022]
Abstract
Sensory neurons are often described in terms of a receptive field, that is, a linear kernel through which stimuli are filtered before they are further processed. If information transmission is assumed to proceed in a feedforward cascade, the receptive field may be interpreted as the external stimulus' profile maximizing neuronal output. The nervous system, however, contains many feedback loops, and sensory neurons filter more currents than the ones representing the transduced external stimulus. Some of the additional currents are generated by the output activity of the neuron itself, and therefore constitute feedback signals. By means of a time-frequency analysis of the input/output transformation, here we show how feedback modifies the receptive field. The model is applicable to various types of feedback processes, from spike-triggered intrinsic conductances to inhibitory synaptic inputs from nearby neurons. We distinguish between the intrinsic receptive field (filtering all input currents) and the effective receptive field (filtering only external stimuli). Whereas the intrinsic receptive field summarizes the biophysical properties of the neuron associated to subthreshold integration and spike generation, only the effective receptive field can be interpreted as the external stimulus' profile maximizing neuronal output. We demonstrate that spike-triggered feedback shifts low-pass filtering towards band-pass processing, transforming integrator neurons into resonators. For strong feedback, a sharp resonance in the spectral neuronal selectivity may appear. Our results provide a unified framework to interpret a collection of previous experimental studies where specific feedback mechanisms were shown to modify the filtering properties of neurons.
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Affiliation(s)
- Eugenio Urdapilleta
- Física Estadística e Interdisciplinaria, Centro Atómico Bariloche, Av. E. Bustillo Km 9.500, S. C. de Bariloche, (8400), Río Negro, Argentina,
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13
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A model of synaptic vesicle-pool depletion and replenishment can account for the interspike interval distributions and nonrenewal properties of spontaneous spike trains of auditory-nerve fibers. J Neurosci 2015; 34:15097-109. [PMID: 25378173 DOI: 10.1523/jneurosci.0903-14.2014] [Citation(s) in RCA: 30] [Impact Index Per Article: 3.3] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 01/27/2023] Open
Abstract
In mammalian auditory systems, the spiking characteristics of each primary afferent (type I auditory-nerve fiber; ANF) are mainly determined by a single ribbon synapse in a single receptor cell (inner hair cell; IHC). ANF spike trains therefore provide a window into the operation of these synapses and cells. It was demonstrated previously (Heil et al., 2007) that the distribution of interspike intervals (ISIs) of cat ANFs during spontaneous activity can be modeled as resulting from refractoriness operating on a non-Poisson stochastic point process of excitation (transmitter release events from the IHC). Here, we investigate nonrenewal properties of these cat-ANF spontaneous spike trains, manifest as negative serial ISI correlations and reduced spike-count variability over short timescales. A previously discussed excitatory process, the constrained failure of events from a homogeneous Poisson point process, can account for these properties, but does not offer a parsimonious explanation for certain trends in the data. We then investigate a three-parameter model of vesicle-pool depletion and replenishment and find that it accounts for all experimental observations, including the ISI distributions, with only the release probability varying between spike trains. The maximum number of units (single vesicles or groups of simultaneously released vesicles) in the readily releasable pool and their replenishment time constant can be assumed to be constant (∼4 and 13.5 ms, respectively). We suggest that the organization of the IHC ribbon synapses not only enables sustained release of neurotransmitter but also imposes temporal regularity on the release process, particularly when operating at high rates.
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14
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Kromer JA, Lindner B, Schimansky-Geier L. Event-triggered feedback in noise-driven phase oscillators. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2014; 89:032138. [PMID: 24730820 DOI: 10.1103/physreve.89.032138] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 11/05/2013] [Indexed: 06/03/2023]
Abstract
Using a stochastic nonlinear phase oscillator model, we study the effect of event-triggered feedback on the statistics of interevent intervals. Events are associated with the entering of a new cycle. The feedback is modeled by an instantaneous increase (positive feedback) or decrease (negative feedback) of the oscillator frequency whenever an event occurs followed by an exponential decay on a slow time scale. In addition to the known excitable and oscillatory regimes, which are separated by a saddle node on invariant circle bifurcation, positive feedback can lead to bistable dynamics and a change of the system's excitability. The feedback has also a strong effect on noise-induced phenomena like coherence resonance or anticoherence resonance. Both positive and negative feedback can lead to more regular output for particular noise strengths. Finally, we investigate serial correlations in the sequence of interevent intervals that occur due to the additional slow dynamics. We derive approximations for the serial correlation coefficient and show that positive feedback results in extended positive interval correlations, whereas negative feedback yields short-ranging negative correlations. Investigating the interplay of feedback and the nonlinear phase dynamics close to the bifurcation, we find that correlations are most pronounced for optimal feedback strengths.
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Affiliation(s)
- Justus A Kromer
- Department of Physics, Humboldt-Universität zu Berlin, Newtonstrasse 15, 12489 Berlin, Germany
| | - Benjamin Lindner
- Department of Physics, Humboldt-Universität zu Berlin, Newtonstrasse 15, 12489 Berlin, Germany and Bernstein Center for Computational Neuroscience Berlin, Berlin, Germany
| | - Lutz Schimansky-Geier
- Department of Physics, Humboldt-Universität zu Berlin, Newtonstrasse 15, 12489 Berlin, Germany and Bernstein Center for Computational Neuroscience Berlin, Berlin, Germany
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15
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Schwalger T, Lindner B. Patterns of interval correlations in neural oscillators with adaptation. Front Comput Neurosci 2013; 7:164. [PMID: 24348372 PMCID: PMC3843362 DOI: 10.3389/fncom.2013.00164] [Citation(s) in RCA: 33] [Impact Index Per Article: 3.0] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 08/01/2013] [Accepted: 10/26/2013] [Indexed: 11/24/2022] Open
Abstract
Neural firing is often subject to negative feedback by adaptation currents. These currents can induce strong correlations among the time intervals between spikes. Here we study analytically the interval correlations of a broad class of noisy neural oscillators with spike-triggered adaptation of arbitrary strength and time scale. Our weak-noise theory provides a general relation between the correlations and the phase-response curve (PRC) of the oscillator, proves anti-correlations between neighboring intervals for adapting neurons with type I PRC and identifies a single order parameter that determines the qualitative pattern of correlations. Monotonically decaying or oscillating correlation structures can be related to qualitatively different voltage traces after spiking, which can be explained by the phase plane geometry. At high firing rates, the long-term variability of the spike train associated with the cumulative interval correlations becomes small, independent of model details. Our results are verified by comparison with stochastic simulations of the exponential, leaky, and generalized integrate-and-fire models with adaptation.
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Affiliation(s)
- Tilo Schwalger
- Bernstein Center for Computational Neuroscience Berlin, Germany ; Department of Physics, Humboldt Universität zu Berlin Berlin, Germany
| | - Benjamin Lindner
- Bernstein Center for Computational Neuroscience Berlin, Germany ; Department of Physics, Humboldt Universität zu Berlin Berlin, Germany
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