1
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Xie M, Muscinelli SP, Decker Harris K, Litwin-Kumar A. Task-dependent optimal representations for cerebellar learning. eLife 2023; 12:e82914. [PMID: 37671785 PMCID: PMC10541175 DOI: 10.7554/elife.82914] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Key Words] [MESH Headings] [Grants] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 08/22/2022] [Accepted: 09/05/2023] [Indexed: 09/07/2023] Open
Abstract
The cerebellar granule cell layer has inspired numerous theoretical models of neural representations that support learned behaviors, beginning with the work of Marr and Albus. In these models, granule cells form a sparse, combinatorial encoding of diverse sensorimotor inputs. Such sparse representations are optimal for learning to discriminate random stimuli. However, recent observations of dense, low-dimensional activity across granule cells have called into question the role of sparse coding in these neurons. Here, we generalize theories of cerebellar learning to determine the optimal granule cell representation for tasks beyond random stimulus discrimination, including continuous input-output transformations as required for smooth motor control. We show that for such tasks, the optimal granule cell representation is substantially denser than predicted by classical theories. Our results provide a general theory of learning in cerebellum-like systems and suggest that optimal cerebellar representations are task-dependent.
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Affiliation(s)
- Marjorie Xie
- Zuckerman Mind Brain Behavior Institute, Columbia UniversityNew YorkUnited States
| | - Samuel P Muscinelli
- Zuckerman Mind Brain Behavior Institute, Columbia UniversityNew YorkUnited States
| | - Kameron Decker Harris
- Department of Computer Science, Western Washington UniversityBellinghamUnited States
| | - Ashok Litwin-Kumar
- Zuckerman Mind Brain Behavior Institute, Columbia UniversityNew YorkUnited States
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2
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Oprea L, Desjardins N, Jiang X, Sareen K, Zheng JQ, Khadra A. Characterizing spontaneous Ca 2+ local transients in OPCs using computational modeling. Biophys J 2022; 121:4419-4432. [PMID: 36352783 PMCID: PMC9748374 DOI: 10.1016/j.bpj.2022.11.007] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [MESH Headings] [Grants] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 06/18/2022] [Revised: 10/03/2022] [Accepted: 11/03/2022] [Indexed: 11/09/2022] Open
Abstract
Spontaneous Ca2+ local transients (SCaLTs) in isolated oligodendrocyte precursor cells are largely regulated by the following fluxes: store-operated Ca2+ entry (SOCE), Na+/Ca2+ exchange, Ca2+ pumping through Ca2+-ATPases, and Ca2+-induced Ca2+-release through ryanodine receptors and inositol-trisphosphate receptors. However, the relative contributions of these fluxes in mediating fast spiking and the slow baseline oscillations seen in SCaLTs remain incompletely understood. Here, we developed a stochastic spatiotemporal computational model to simulate SCaLTs in a homogeneous medium with ionic flow between the extracellular, cytoplasmic, and endoplasmic-reticulum compartments. By simulating the model and plotting both the histograms of SCaLTs obtained experimentally and from the model as well as the standard deviation of inter-SCaLT intervals against inter-SCaLT interval averages of multiple model and experimental realizations, we revealed the following: (1) SCaLTs exhibit very similar characteristics between the two data sets, (2) they are mostly random, (3) they encode information in their frequency, and (4) their slow baseline oscillations could be due to the stochastic slow clustering of inositol-trisphosphate receptors (modeled as an Ornstein-Uhlenbeck noise process). Bifurcation analysis of a deterministic temporal version of the model showed that the contribution of fluxes to SCaLTs depends on the parameter regime and that the combination of excitability, stochasticity, and mixed-mode oscillations are responsible for irregular spiking and doublets in SCaLTs. Additionally, our results demonstrated that blocking each flux reduces SCaLTs' frequency and that the reverse (forward) mode of Na+/Ca2+ exchange decreases (increases) SCaLTs. Taken together, these results provide a quantitative framework for SCaLT formation in oligodendrocyte precursor cells.
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Affiliation(s)
- Lawrence Oprea
- Department of Physiology, McGill University, Montréal, Quebec, Canada
| | | | - Xiaoyu Jiang
- Department of Physiology, McGill University, Montréal, Quebec, Canada
| | - Kushagra Sareen
- Department of Physiology, McGill University, Montréal, Quebec, Canada
| | - James Q Zheng
- Department of Cell Biology, School of Medicine, Emory University, Atlanta, Georgia
| | - Anmar Khadra
- Department of Physiology, McGill University, Montréal, Quebec, Canada.
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3
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Cunha GM, Corso G, Miranda JGV, Dos Santos Lima GZ. Ephaptic entrainment in hybrid neuronal model. Sci Rep 2022; 12:1629. [PMID: 35102158 PMCID: PMC8803837 DOI: 10.1038/s41598-022-05343-3] [Citation(s) in RCA: 6] [Impact Index Per Article: 3.0] [Reference Citation Analysis] [Abstract] [Key Words] [MESH Headings] [Grants] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 07/22/2021] [Accepted: 01/04/2022] [Indexed: 11/17/2022] Open
Abstract
In recent decades, there has been a growing interest in the impact of electric fields generated in the brain. Transmembrane ionic currents originate electric fields in the extracellular space and are capable of affecting nearby neurons, a phenomenon called ephaptic neuronal communication. In the present work, the Quadratic Integrated-and-Fire model (QIF-E) underwent an adjustment/improvement to include the ephaptic entrainment behavior between neurons and electric fields. Indeed, the aim of our study is to validate the QIF-E model, which is a model to estimate the influence of electric fields on neurons. For this purpose, we evaluated whether the main properties observed in an experiment by Anastassiou et al. (Nat Neurosci 14:217-223, 2011), which analyzed the effect of an electric field on cortical pyramidal neurons, are reproduced with the QIF-E model. In this way, the analysis tools are employed according to the neuronal activity regime: (i) for the subthreshold regime, the circular statistic is used to describe the phase differences between the input stimulus signal (electrode) and the modeled membrane response; (ii) in the suprathreshold regime, the Population Vector and the Spike Field Coherence are used to estimate phase preferences and the entrainment intensity between the input stimulus and Action Potentials. The results observed are (i) in the subthreshold regime the values of the phase differences change with distinct frequencies of the input stimulus; (ii) in the supra-threshold regime the preferential phase of Action Potentials changes for different frequencies. In addition, we explore other parameters of the model, such as noise and membrane characteristic-time, in order to understand different types of neurons and extracellular environment related to ephaptic communication. Such results are consistent with results observed in empirical experiments based on ephaptic phenomenon. In addition, the QIF-E model allows further studies on the physiological importance of ephaptic communication in the brain, and its simplicity may open a door to simulate the ephaptic response in neuronal networks and assess the impact of ephaptic communication in such scenarios.
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Affiliation(s)
- Gabriel Moreno Cunha
- Departamento de Física Teórica e Experimental, Universidade Federal do Rio Grande do Norte, Natal, RN, 59078-970, Brazil
| | - Gilberto Corso
- Departamento de Física Teórica e Experimental, Universidade Federal do Rio Grande do Norte, Natal, RN, 59078-970, Brazil
- Departamento de Biofísica e Farmacologia, Universidade Federal do Rio Grande do Norte, Natal, RN, 59078-970, Brazil
| | | | - Gustavo Zampier Dos Santos Lima
- Departamento de Física Teórica e Experimental, Universidade Federal do Rio Grande do Norte, Natal, RN, 59078-970, Brazil.
- Escola de Ciências e Tecnologia, Universidade Federal do Rio Grande do Norte, Natal, RN, 59078-970, Brazil.
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4
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Zagkos L, Roberts J, Auley MM. A mathematical model which examines age-related stochastic fluctuations in DNA maintenance methylation. Exp Gerontol 2021; 156:111623. [PMID: 34774717 DOI: 10.1016/j.exger.2021.111623] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 06/14/2021] [Revised: 10/30/2021] [Accepted: 11/04/2021] [Indexed: 10/19/2022]
Abstract
Due to its complexity and its ubiquitous nature the ageing process remains an enduring biological puzzle. Many molecular mechanisms and biochemical process have become synonymous with ageing. However, recent findings have pinpointed epigenetics as having a key role in ageing and healthspan. In particular age related changes to DNA methylation offer the possibility of monitoring the trajectory of biological ageing and could even be used to predict the onset of diseases such as cancer, Alzheimer's disease and cardiovascular disease. At the molecular level emerging evidence strongly suggests the regulatory processes which govern DNA methylation are subject to intracellular stochasticity. It is challenging to fully understand the impact of stochasticity on DNA methylation levels at the molecular level experimentally. An ideal solution is to use mathematical models to capture the essence of the stochasticity and its outcomes. In this paper we present a novel stochastic model which accounts for specific methylation levels within a gene promoter. Uncertainty of the eventual site-specific methylation levels for different values of methylation age, depending on the initial methylation levels were analysed. Our model predicts the observed bistable levels in CpG islands. In addition, simulations with various levels of noise indicate that uncertainty predominantly spreads through the hypermethylated region of stability, especially for large values of input noise. A key outcome of the model is that CpG islands with high to intermediate methylation levels tend to be more susceptible to dramatic DNA methylation changes due to increasing methylation age.
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Affiliation(s)
- Loukas Zagkos
- Department of Mathematics, School of Science and Engineering, University of Chester, Thornton Science Park, Chester CH2 4NU, UK; Department of Epidemiology and Biostatistics, School of Public Health, Imperial College London, St Mary's Campus, London W2 1PG, UK.
| | - Jason Roberts
- Department of Mathematics, School of Science and Engineering, University of Chester, Thornton Science Park, Chester CH2 4NU, UK
| | - Mark Mc Auley
- Department of Chemical Engineering, School of Science and Engineering, University of Chester, Thornton Science Park, Chester CH2 4NU, UK
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5
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Rongala UB, Enander JMD, Kohler M, Loeb GE, Jörntell H. A Non-spiking Neuron Model With Dynamic Leak to Avoid Instability in Recurrent Networks. Front Comput Neurosci 2021; 15:656401. [PMID: 34093156 PMCID: PMC8173185 DOI: 10.3389/fncom.2021.656401] [Citation(s) in RCA: 3] [Impact Index Per Article: 1.0] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 01/20/2021] [Accepted: 04/19/2021] [Indexed: 11/18/2022] Open
Abstract
Recurrent circuitry components are distributed widely within the brain, including both excitatory and inhibitory synaptic connections. Recurrent neuronal networks have potential stability problems, perhaps a predisposition to epilepsy. More generally, instability risks making internal representations of information unreliable. To assess the inherent stability properties of such recurrent networks, we tested a linear summation, non-spiking neuron model with and without a “dynamic leak”, corresponding to the low-pass filtering of synaptic input current by the RC circuit of the biological membrane. We first show that the output of this neuron model, in either of its two forms, follows its input at a higher fidelity than a wide range of spiking neuron models across a range of input frequencies. Then we constructed fully connected recurrent networks with equal numbers of excitatory and inhibitory neurons and randomly distributed weights across all synapses. When the networks were driven by pseudorandom sensory inputs with varying frequency, the recurrent network activity tended to induce high frequency self-amplifying components, sometimes evident as distinct transients, which were not present in the input data. The addition of a dynamic leak based on known membrane properties consistently removed such spurious high frequency noise across all networks. Furthermore, we found that the neuron model with dynamic leak imparts a network stability that seamlessly scales with the size of the network, conduction delays, the input density of the sensory signal and a wide range of synaptic weight distributions. Our findings suggest that neuronal dynamic leak serves the beneficial function of protecting recurrent neuronal circuitry from the self-induction of spurious high frequency signals, thereby permitting the brain to utilize this architectural circuitry component regardless of network size or recurrency.
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Affiliation(s)
- Udaya B Rongala
- Department of Experimental Medical Science, Faculty of Medicine, Lund University, Lund, Sweden
| | - Jonas M D Enander
- Department of Experimental Medical Science, Faculty of Medicine, Lund University, Lund, Sweden
| | - Matthias Kohler
- Department of Informatics, Technical University of Munich, Munich, Germany
| | - Gerald E Loeb
- Department of Biomedical Engineering, Viterbi School of Engineering, University of Southern California, Los Angeles, CA, United States
| | - Henrik Jörntell
- Department of Experimental Medical Science, Faculty of Medicine, Lund University, Lund, Sweden
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6
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Rongala UB, Mazzoni A, Spanne A, Jörntell H, Oddo CM. Cuneate spiking neural network learning to classify naturalistic texture stimuli under varying sensing conditions. Neural Netw 2020; 123:273-287. [PMID: 31887687 DOI: 10.1016/j.neunet.2019.11.020] [Citation(s) in RCA: 6] [Impact Index Per Article: 1.5] [Reference Citation Analysis] [Abstract] [Key Words] [MESH Headings] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 03/15/2019] [Revised: 10/22/2019] [Accepted: 11/25/2019] [Indexed: 11/18/2022]
Abstract
We implemented a functional neuronal network that was able to learn and discriminate haptic features from biomimetic tactile sensor inputs using a two-layer spiking neuron model and homeostatic synaptic learning mechanism. The first order neuron model was used to emulate biological tactile afferents and the second order neuron model was used to emulate biological cuneate neurons. We have evaluated 10 naturalistic textures using a passive touch protocol, under varying sensing conditions. Tactile sensor data acquired with five textures under five sensing conditions were used for a synaptic learning process, to tune the synaptic weights between tactile afferents and cuneate neurons. Using post-learning synaptic weights, we evaluated the individual and population cuneate neuron responses by decoding across 10 stimuli, under varying sensing conditions. This resulted in a high decoding performance. We further validated the decoding performance across stimuli, irrespective of sensing velocities using a set of 25 cuneate neuron responses. This resulted in a median decoding performance of 96% across the set of cuneate neurons. Being able to learn and perform generalized discrimination across tactile stimuli, makes this functional spiking tactile system effective and suitable for further robotic applications.
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Affiliation(s)
- Udaya B Rongala
- The BioRobotics Institute, Sant'Anna School of Advanced Studies, 56127 Pisa, Italy; Department of Linguistics and Comparative Cultural Studies, Ca' Foscari University of Venice, 30123 Venice, Italy; Neural Basis of Sensorimotor Control, Department of Experimental Medical Science, Lund University, SE-221 84 Lund, Sweden; Department of Excellence in Robotics & AI, Scuola Superiore Sant'Anna, Pisa, Italy.
| | - Alberto Mazzoni
- The BioRobotics Institute, Sant'Anna School of Advanced Studies, 56127 Pisa, Italy; Department of Excellence in Robotics & AI, Scuola Superiore Sant'Anna, Pisa, Italy
| | - Anton Spanne
- Neural Basis of Sensorimotor Control, Department of Experimental Medical Science, Lund University, SE-221 84 Lund, Sweden
| | - Henrik Jörntell
- Neural Basis of Sensorimotor Control, Department of Experimental Medical Science, Lund University, SE-221 84 Lund, Sweden
| | - Calogero M Oddo
- The BioRobotics Institute, Sant'Anna School of Advanced Studies, 56127 Pisa, Italy; Department of Excellence in Robotics & AI, Scuola Superiore Sant'Anna, Pisa, Italy.
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7
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Rongala UB, Spanne A, Mazzoni A, Bengtsson F, Oddo CM, Jörntell H. Intracellular Dynamics in Cuneate Nucleus Neurons Support Self-Stabilizing Learning of Generalizable Tactile Representations. Front Cell Neurosci 2018; 12:210. [PMID: 30108485 PMCID: PMC6079306 DOI: 10.3389/fncel.2018.00210] [Citation(s) in RCA: 23] [Impact Index Per Article: 3.8] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 03/26/2018] [Accepted: 06/26/2018] [Indexed: 12/12/2022] Open
Abstract
How the brain represents the external world is an unresolved issue for neuroscience, which could provide fundamental insights into brain circuitry operation and solutions for artificial intelligence and robotics. The neurons of the cuneate nucleus form the first interface for the sense of touch in the brain. They were previously shown to have a highly skewed synaptic weight distribution for tactile primary afferent inputs, suggesting that their connectivity is strongly shaped by learning. Here we first characterized the intracellular dynamics and inhibitory synaptic inputs of cuneate neurons in vivo and modeled their integration of tactile sensory inputs. We then replaced the tactile inputs with input from a sensorized bionic fingertip and modeled the learning-induced representations that emerged from varied sensory experiences. The model reproduced both the intrinsic membrane dynamics and the synaptic weight distributions observed in cuneate neurons in vivo. In terms of higher level model properties, individual cuneate neurons learnt to identify specific sets of correlated sensors, which at the population level resulted in a decomposition of the sensor space into its recurring high-dimensional components. Such vector components could be applied to identify both past and novel sensory experiences and likely correspond to the fundamental haptic input features these neurons encode in vivo. In addition, we show that the cuneate learning architecture is robust to a wide range of intrinsic parameter settings due to the neuronal intrinsic dynamics. Therefore, the architecture is a potentially generic solution for forming versatile representations of the external world in different sensor systems.
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Affiliation(s)
- Udaya B Rongala
- The BioRobotics Institute, Scuola Superiore Sant'Anna, Pisa, Italy
| | - Anton Spanne
- Section for Neurobiology, Department of Experimental Medical Sciences, Biomedical Center, Lund University, Lund, Sweden
| | - Alberto Mazzoni
- The BioRobotics Institute, Scuola Superiore Sant'Anna, Pisa, Italy
| | - Fredrik Bengtsson
- Section for Neurobiology, Department of Experimental Medical Sciences, Biomedical Center, Lund University, Lund, Sweden
| | - Calogero M Oddo
- The BioRobotics Institute, Scuola Superiore Sant'Anna, Pisa, Italy
| | - Henrik Jörntell
- Section for Neurobiology, Department of Experimental Medical Sciences, Biomedical Center, Lund University, Lund, Sweden
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8
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Abstract
The model studied in this paper is a stochastic extension of the so-called neuron model introduced by Hodgkin and Huxley. In the sense of rough paths, the model is perturbed by a multiplicative noise driven by a fractional Brownian motion, with a vector field satisfying the viability condition of Coutin and Marie for [Formula: see text]. An application to the modeling of the membrane potential of nerve fibers damaged by a neuropathy is provided.
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Affiliation(s)
- Laure Coutin
- Institut de Mathématiques de Toulouse, Toulouse, France
| | | | - Nicolas Marie
- Laboratoire Modal’X, Université Paris 10, Nanterre, France
- ESME Sudria, Paris, France
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9
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DeLeon O, Hodis H, O’Malley Y, Johnson J, Salimi H, Zhai Y, Winter E, Remec C, Eichelberger N, Van Cleave B, Puliadi R, Harrington RD, Stapleton JT, Haim H. Accurate predictions of population-level changes in sequence and structural properties of HIV-1 Env using a volatility-controlled diffusion model. PLoS Biol 2017; 15:e2001549. [PMID: 28384158 PMCID: PMC5383018 DOI: 10.1371/journal.pbio.2001549] [Citation(s) in RCA: 19] [Impact Index Per Article: 2.7] [Reference Citation Analysis] [Abstract] [MESH Headings] [Grants] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 11/11/2016] [Accepted: 03/06/2017] [Indexed: 01/08/2023] Open
Abstract
The envelope glycoproteins (Envs) of HIV-1 continuously evolve in the host by random mutations and recombination events. The resulting diversity of Env variants circulating in the population and their continuing diversification process limit the efficacy of AIDS vaccines. We examined the historic changes in Env sequence and structural features (measured by integrity of epitopes on the Env trimer) in a geographically defined population in the United States. As expected, many Env features were relatively conserved during the 1980s. From this state, some features diversified whereas others remained conserved across the years. We sought to identify “clues” to predict the observed historic diversification patterns. Comparison of viruses that cocirculate in patients at any given time revealed that each feature of Env (sequence or structural) exists at a defined level of variance. The in-host variance of each feature is highly conserved among individuals but can vary between different HIV-1 clades. We designate this property “volatility” and apply it to model evolution of features as a linear diffusion process that progresses with increasing genetic distance. Volatilities of different features are highly correlated with their divergence in longitudinally monitored patients. Volatilities of features also correlate highly with their population-level diversification. Using volatility indices measured from a small number of patient samples, we accurately predict the population diversity that developed for each feature over the course of 30 years. Amino acid variants that evolved at key antigenic sites are also predicted well. Therefore, small “fluctuations” in feature values measured in isolated patient samples accurately describe their potential for population-level diversification. These tools will likely contribute to the design of population-targeted AIDS vaccines by effectively capturing the diversity of currently circulating strains and addressing properties of variants expected to appear in the future. HIV-1 is the causative agent of the global AIDS pandemic. The envelope glycoproteins (Envs) of HIV-1 constitute a primary target for antibody-based vaccines. However, the diversity of Envs in the population limits the potential efficacy of this approach. Accurate estimates of the range of variants that currently infect patients and those expected to appear in the future will likely contribute to the design of population-targeted immunogens. We found that different properties (features) of Env have different propensities for small “fluctuations” in their values among viruses that infect patients at any given time point. This propensity of each feature for in-host variance, which we designate “volatility”, is conserved among patients. We apply this parameter to model the evolution of features (in patients and population) as a diffusion process driven by their “diffusion coefficients” (volatilities). Using volatilities measured from a few patient samples from the 1980s, we accurately predict properties of viruses that evolved in the population over the course of 30 years. The diffusion-based model described here efficiently captures evolution of phenotypes in biological systems controlled by a dominant random component.
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Affiliation(s)
- Orlando DeLeon
- Department of Microbiology, Carver College of Medicine, University of Iowa, Iowa City, Iowa, United States of America
| | - Hagit Hodis
- Department of Microbiology, Carver College of Medicine, University of Iowa, Iowa City, Iowa, United States of America
| | - Yunxia O’Malley
- Department of Microbiology, Carver College of Medicine, University of Iowa, Iowa City, Iowa, United States of America
| | - Jacklyn Johnson
- Department of Microbiology, Carver College of Medicine, University of Iowa, Iowa City, Iowa, United States of America
| | - Hamid Salimi
- Department of Microbiology, Carver College of Medicine, University of Iowa, Iowa City, Iowa, United States of America
| | - Yinjie Zhai
- Department of Microbiology, Carver College of Medicine, University of Iowa, Iowa City, Iowa, United States of America
| | - Elizabeth Winter
- Department of Microbiology, Carver College of Medicine, University of Iowa, Iowa City, Iowa, United States of America
| | - Claire Remec
- Department of Microbiology, Carver College of Medicine, University of Iowa, Iowa City, Iowa, United States of America
| | - Noah Eichelberger
- Department of Microbiology, Carver College of Medicine, University of Iowa, Iowa City, Iowa, United States of America
| | - Brandon Van Cleave
- Department of Microbiology, Carver College of Medicine, University of Iowa, Iowa City, Iowa, United States of America
| | - Ramya Puliadi
- Department of Microbiology, Carver College of Medicine, University of Iowa, Iowa City, Iowa, United States of America
| | - Robert D. Harrington
- Center for AIDS Research (CFAR) at the University of Washington, Seattle, Washington, United States of America
| | - Jack T. Stapleton
- Department of Microbiology, Carver College of Medicine, University of Iowa, Iowa City, Iowa, United States of America
- Veterans Affairs Medical Center, Iowa City, Iowa, United States of America
| | - Hillel Haim
- Department of Microbiology, Carver College of Medicine, University of Iowa, Iowa City, Iowa, United States of America
- * E-mail:
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10
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Maisel B, Lindenberg K. Channel noise effects on first spike latency of a stochastic Hodgkin-Huxley neuron. Phys Rev E 2017; 95:022414. [PMID: 28297877 DOI: 10.1103/physreve.95.022414] [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: 09/06/2016] [Indexed: 06/06/2023]
Abstract
While it is widely accepted that information is encoded in neurons via action potentials or spikes, it is far less understood what specific features of spiking contain encoded information. Experimental evidence has suggested that the timing of the first spike may be an energy-efficient coding mechanism that contains more neural information than subsequent spikes. Therefore, the biophysical features of neurons that underlie response latency are of considerable interest. Here we examine the effects of channel noise on the first spike latency of a Hodgkin-Huxley neuron receiving random input from many other neurons. Because the principal feature of a Hodgkin-Huxley neuron is the stochastic opening and closing of channels, the fluctuations in the number of open channels lead to fluctuations in the membrane voltage and modify the timing of the first spike. Our results show that when a neuron has a larger number of channels, (i) the occurrence of the first spike is delayed and (ii) the variation in the first spike timing is greater. We also show that the mean, median, and interquartile range of first spike latency can be accurately predicted from a simple linear regression by knowing only the number of channels in the neuron and the rate at which presynaptic neurons fire, but the standard deviation (i.e., neuronal jitter) cannot be predicted using only this information. We then compare our results to another commonly used stochastic Hodgkin-Huxley model and show that the more commonly used model overstates the first spike latency but can predict the standard deviation of first spike latencies accurately. We end by suggesting a more suitable definition for the neuronal jitter based upon our simulations and comparison of the two models.
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Affiliation(s)
- Brenton Maisel
- Department of Chemistry and Biochemistry, and BioCircuits Institute, University of California San Diego, La Jolla, California 92093-0340, USA
| | - Katja Lindenberg
- Department of Chemistry and Biochemistry, and BioCircuits Institute, University of California San Diego, La Jolla, California 92093-0340, USA
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11
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Requena-Carrión J, Requena-Carrión VJ. Distribution of transition times in a stochastic model of excitable cell: Insights into the cell-intrinsic mechanisms of randomness in neuronal interspike intervals. Phys Rev E 2016; 93:042418. [PMID: 27176339 DOI: 10.1103/physreve.93.042418] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 06/09/2015] [Indexed: 11/07/2022]
Abstract
In this paper, we develop an analytical approach to studying random patterns of activity in excitable cells. Our analytical approach uses a two-state stochastic model of excitable system based on the electrophysiological properties of refractoriness and restitution, which characterize cell recovery after excitation. By applying the notion of probability density flux, we derive the distributions of transition times between states and the distribution of interspike interval (ISI) durations for a constant applied stimulus. The derived ISI distribution is unimodal and, provided that the time spent in the excited state is constant, can be approximated by a Rayleigh peak followed by an exponential tail. We then explore the role of the model parameters in determining the shape of the derived distributions and the ISI coefficient of variation. Finally, we use our analytical results to study simulation results from the stochastic Morris-Lecar neuron and from a three-state extension of the proposed stochastic model, which is capable of reproducing multimodal ISI histograms.
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12
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Ling A, Huang Y, Shuai J, Lan Y. Channel based generating function approach to the stochastic Hodgkin-Huxley neuronal system. Sci Rep 2016; 6:22662. [PMID: 26940002 PMCID: PMC4778126 DOI: 10.1038/srep22662] [Citation(s) in RCA: 4] [Impact Index Per Article: 0.5] [Reference Citation Analysis] [Abstract] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 12/08/2015] [Accepted: 02/09/2016] [Indexed: 11/09/2022] Open
Abstract
Internal and external fluctuations, such as channel noise and synaptic noise, contribute to the generation of spontaneous action potentials in neurons. Many different Langevin approaches have been proposed to speed up the computation but with waning accuracy especially at small channel numbers. We apply a generating function approach to the master equation for the ion channel dynamics and further propose two accelerating algorithms, with an accuracy close to the Gillespie algorithm but with much higher efficiency, opening the door for expedited simulation of noisy action potential propagating along axons or other types of noisy signal transduction.
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Affiliation(s)
- Anqi Ling
- Department of Physics, Tsinghua University, Beijing 100084, China.,Collaborative Innovation Center of Quantum Matter, Beijing 100084, China
| | - Yandong Huang
- Department of Physics and Institute of Theoretical Physics and Astrophysics, Xiamen University, Xiamen 361005, China
| | - Jianwei Shuai
- Department of Physics and Institute of Theoretical Physics and Astrophysics, Xiamen University, Xiamen 361005, China
| | - Yueheng Lan
- Department of Physics, Tsinghua University, Beijing 100084, China.,Collaborative Innovation Center of Quantum Matter, Beijing 100084, China
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13
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Mei Y, Carbo A, Hoops S, Hontecillas R, Bassaganya-Riera J. ENISI SDE: A New Web-Based Tool for Modeling Stochastic Processes. IEEE/ACM TRANSACTIONS ON COMPUTATIONAL BIOLOGY AND BIOINFORMATICS 2015; 12:289-297. [PMID: 26357217 DOI: 10.1109/tcbb.2014.2351823] [Citation(s) in RCA: 3] [Impact Index Per Article: 0.3] [Reference Citation Analysis] [Abstract] [MESH Headings] [Grants] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 06/05/2023]
Abstract
Modeling and simulations approaches have been widely used in computational biology, mathematics, bioinformatics and engineering to represent complex existing knowledge and to effectively generate novel hypotheses. While deterministic modeling strategies are widely used in computational biology, stochastic modeling techniques are not as popular due to a lack of user-friendly tools. This paper presents ENISI SDE, a novel web-based modeling tool with stochastic differential equations. ENISI SDE provides user-friendly web user interfaces to facilitate adoption by immunologists and computational biologists. This work provides three major contributions: (1) discussion of SDE as a generic approach for stochastic modeling in computational biology; (2) development of ENISI SDE, a web-based user-friendly SDE modeling tool that highly resembles regular ODE-based modeling; (3) applying ENISI SDE modeling tool through a use case for studying stochastic sources of cell heterogeneity in the context of CD4+ T cell differentiation. The CD4+ T cell differential ODE model has been published [8] and can be downloaded from biomodels.net. The case study reproduces a biological phenomenon that is not captured by the previously published ODE model and shows the effectiveness of SDE as a stochastic modeling approach in biology in general and immunology in particular and the power of ENISI SDE.
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Spanne A, Geborek P, Bengtsson F, Jörntell H. Simulating spinal border cells and cerebellar granule cells under locomotion--a case study of spinocerebellar information processing. PLoS One 2014; 9:e107793. [PMID: 25226298 PMCID: PMC4166671 DOI: 10.1371/journal.pone.0107793] [Citation(s) in RCA: 9] [Impact Index Per Article: 0.9] [Reference Citation Analysis] [Abstract] [MESH Headings] [Grants] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 06/26/2013] [Accepted: 08/23/2014] [Indexed: 11/18/2022] Open
Abstract
The spinocerebellar systems are essential for the brain in the performance of coordinated movements, but our knowledge about the spinocerebellar interactions is very limited. Recently, several crucial pieces of information have been acquired for the spinal border cell (SBC) component of the ventral spinocerebellar tract (VSCT), as well as the effects of SBC mossy fiber activation in granule cells of the cerebellar cortex. SBCs receive monosynaptic input from the reticulospinal tract (RST), which is an important driving system under locomotion, and disynaptic inhibition from Ib muscle afferents. The patterns of activity of RST neurons and Ib afferents under locomotion are known. The activity of VSCT neurons under fictive locomotion, i.e. without sensory feedback, is also known, but there is little information on how these neurons behave under actual locomotion and for cerebellar granule cells receiving SBC input this is completely unknown. But the available information makes it possible to simulate the interactions between the spinal and cerebellar neuronal circuitries with a relatively large set of biological constraints. Using a model of the various neuronal elements and the network they compose, we simulated the modulation of the SBCs and their target granule cells under locomotion and hence generated testable predictions of their general pattern of modulation under this condition. This particular system offers a unique opportunity to simulate these interactions with a limited number of assumptions, which helps making the model biologically plausible. Similar principles of information processing may be expected to apply to all spinocerebellar systems.
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Affiliation(s)
- Anton Spanne
- Neural Basis of Sensorimotor Control, Department of Experimental Medical Science, Lund University, Lund, Sweden
| | - Pontus Geborek
- Neural Basis of Sensorimotor Control, Department of Experimental Medical Science, Lund University, Lund, Sweden
| | - Fredrik Bengtsson
- Neural Basis of Sensorimotor Control, Department of Experimental Medical Science, Lund University, Lund, Sweden
| | - Henrik Jörntell
- Neural Basis of Sensorimotor Control, Department of Experimental Medical Science, Lund University, Lund, Sweden
- * E-mail:
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Spanne A, Geborek P, Bengtsson F, Jörntell H. Spike generation estimated from stationary spike trains in a variety of neurons in vivo. Front Cell Neurosci 2014; 8:199. [PMID: 25120429 PMCID: PMC4111083 DOI: 10.3389/fncel.2014.00199] [Citation(s) in RCA: 23] [Impact Index Per Article: 2.3] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 04/25/2014] [Accepted: 07/02/2014] [Indexed: 12/03/2022] Open
Abstract
To any model of brain function, the variability of neuronal spike firing is a problem that needs to be taken into account. Whereas the synaptic integration can be described in terms of the original Hodgkin-Huxley (H-H) formulations of conductance-based electrical signaling, the transformation of the resulting membrane potential into patterns of spike output is subjected to stochasticity that may not be captured with standard single neuron H-H models. The dynamics of the spike output is dependent on the normal background synaptic noise present in vivo, but the neuronal spike firing variability in vivo is not well studied. In the present study, we made long-term whole cell patch clamp recordings of stationary spike firing states across a range of membrane potentials from a variety of subcortical neurons in the non-anesthetized, decerebrated state in vivo. Based on the data, we formulated a simple, phenomenological model of the properties of the spike generation in each neuron that accurately captured the stationary spike firing statistics across all membrane potentials. The model consists of a parametric relationship between the mean and standard deviation of the inter-spike intervals, where the parameter is linearly related to the injected current over the membrane. This enabled it to generate accurate approximations of spike firing also under inhomogeneous conditions with input that varies over time. The parameters describing the spike firing statistics for different neuron types overlapped extensively, suggesting that the spike generation had similar properties across neurons.
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Affiliation(s)
- Anton Spanne
- Neural Basis of Sensorimotor Control, Department of Experimental Medical Science, Lund University Lund, Sweden
| | - Pontus Geborek
- Neural Basis of Sensorimotor Control, Department of Experimental Medical Science, Lund University Lund, Sweden
| | - Fredrik Bengtsson
- Neural Basis of Sensorimotor Control, Department of Experimental Medical Science, Lund University Lund, Sweden
| | - Henrik Jörntell
- Neural Basis of Sensorimotor Control, Department of Experimental Medical Science, Lund University Lund, Sweden
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Tuckwell HC, Penington NJ. Computational modeling of spike generation in serotonergic neurons of the dorsal raphe nucleus. Prog Neurobiol 2014; 118:59-101. [PMID: 24784445 DOI: 10.1016/j.pneurobio.2014.04.001] [Citation(s) in RCA: 12] [Impact Index Per Article: 1.2] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 10/11/2012] [Revised: 04/14/2014] [Accepted: 04/21/2014] [Indexed: 01/14/2023]
Abstract
Serotonergic neurons of the dorsal raphe nucleus, with their extensive innervation of limbic and higher brain regions and interactions with the endocrine system have important modulatory or regulatory effects on many cognitive, emotional and physiological processes. They have been strongly implicated in responses to stress and in the occurrence of major depressive disorder and other psychiatric disorders. In order to quantify some of these effects, detailed mathematical models of the activity of such cells are required which describe their complex neurochemistry and neurophysiology. We consider here a single-compartment model of these neurons which is capable of describing many of the known features of spike generation, particularly the slow rhythmic pacemaking activity often observed in these cells in a variety of species. Included in the model are 11 kinds of ion channels: a fast sodium current INa, a delayed rectifier potassium current IKDR, a transient potassium current IA, a slow non-inactivating potassium current IM, a low-threshold calcium current IT, two high threshold calcium currents IL and IN, small and large conductance potassium currents ISK and IBK, a hyperpolarization-activated cation current IH and a leak current ILeak. In Sections 3-8, each current type is considered in detail and parameters estimated from voltage clamp data where possible. Three kinds of model are considered for the BK current and two for the leak current. Intracellular calcium ion concentration Cai is an additional component and calcium dynamics along with buffering and pumping is discussed in Section 9. The remainder of the article contains descriptions of computed solutions which reveal both spontaneous and driven spiking with several parameter sets. Attention is focused on the properties usually associated with these neurons, particularly long duration of action potential, steep upslope on the leading edge of spikes, pacemaker-like spiking, long-lasting afterhyperpolarization and the ramp-like return to threshold after a spike. In some cases the membrane potential trajectories display doublets or have humps or notches as have been reported in some experimental studies. The computed time courses of IA and IT during the interspike interval support the generally held view of a competition between them in influencing the frequency of spiking. Spontaneous activity was facilitated by the presence of IH which has been found in these neurons by some investigators. For reasonable sets of parameters spike frequencies between about 0.6Hz and 1.2Hz are obtained, but frequencies as high as 6Hz could be obtained with special parameter choices. Topics investigated and compared with experiment include shoulders, notches, anodal break phenomena, the effects of noradrenergic input, frequency versus current curves, depolarization block, effects of cell size and the effects of IM. The inhibitory effects of activating 5-HT1A autoreceptors are also investigated. There is a considerable discussion of in vitro versus in vivo firing behavior, with focus on the roles of noradrenergic input, corticotropin-releasing factor and orexinergic inputs. Location of cells within the nucleus is probably a major factor, along with the state of the animal.
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Affiliation(s)
- Henry C Tuckwell
- Max Planck Institute for Mathematics in the Sciences, Inselstr. 22, 04103 Leipzig, Germany; School of Electrical and Electronic Engineering, University of Adelaide, Adelaide, South Australia 5005, Australia.
| | - Nicholas J Penington
- Department of Physiology and Pharmacology, State University of New York, Downstate Medical Center, Box 29, 450 Clarkson Avenue, Brooklyn, NY 11203-2098, USA; Program in Neural and Behavioral Science and Robert F. Furchgott Center for Neural and Behavioral Science, State University of New York, Downstate Medical Center, Box 29, 450 Clarkson Avenue, Brooklyn, NY 11203-2098, USA
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Linne ML, Jalonen TO. Astrocyte-neuron interactions: from experimental research-based models to translational medicine. PROGRESS IN MOLECULAR BIOLOGY AND TRANSLATIONAL SCIENCE 2014; 123:191-217. [PMID: 24560146 DOI: 10.1016/b978-0-12-397897-4.00005-x] [Citation(s) in RCA: 17] [Impact Index Per Article: 1.7] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 12/19/2022]
Abstract
In this chapter, we review the principal astrocyte functions and the interactions between neurons and astrocytes. We then address how the experimentally observed functions have been verified in computational models and review recent experimental literature on astrocyte-neuron interactions. Benefits of computational neuroscience work are highlighted through selected studies with neurons and astrocytes by analyzing the existing models qualitatively and assessing the relevance of these models to experimental data. Common strategies to mathematical modeling and computer simulation in neuroscience are summarized for the nontechnical reader. The astrocyte-neuron interactions are then further illustrated by examples of some neurological and neurodegenerative diseases, where the miscommunication between glia and neurons is found to be increasingly important.
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Affiliation(s)
- Marja-Leena Linne
- Computational Neuroscience Group, Department of Signal Processing, Tampere University of Technology, Tampere, Finland
| | - Tuula O Jalonen
- Department of Physiology and Neuroscience, St. George's University, School of Medicine, Grenada, West Indies
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Comparison of models for IP3 receptor kinetics using stochastic simulations. PLoS One 2013; 8:e59618. [PMID: 23630568 PMCID: PMC3629942 DOI: 10.1371/journal.pone.0059618] [Citation(s) in RCA: 8] [Impact Index Per Article: 0.7] [Reference Citation Analysis] [Abstract] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 11/14/2012] [Accepted: 02/15/2013] [Indexed: 12/07/2022] Open
Abstract
Inositol 1,4,5-trisphosphate receptor (IP3R) is a ubiquitous intracellular calcium (Ca2+) channel which has a major role in controlling Ca2+ levels in neurons. A variety of computational models have been developed to describe the kinetic function of IP3R under different conditions. In the field of computational neuroscience, it is of great interest to apply the existing models of IP3R when modeling local Ca2+ transients in dendrites or overall Ca2+ dynamics in large neuronal models. The goal of this study was to evaluate existing IP3R models, based on electrophysiological data. This was done in order to be able to suggest suitable models for neuronal modeling. Altogether four models (Othmer and Tang, 1993; Dawson etal., 2003; Fraiman and Dawson, 2004; Doi etal., 2005) were selected for a more detailed comparison. The selection was based on the computational efficiency of the models and the type of experimental data that was used in developing the model. The kinetics of all four models were simulated by stochastic means, using the simulation software STEPS, which implements the Gillespie stochastic simulation algorithm. The results show major differences in the statistical properties of model functionality. Of the four compared models, the one by Fraiman and Dawson (2004) proved most satisfactory in producing the specific features of experimental findings reported in literature. To our knowledge, the present study is the first detailed evaluation of IP3R models using stochastic simulation methods, thus providing an important setting for constructing a new, realistic model of IP3R channel kinetics for compartmental modeling of neuronal functions. We conclude that the kinetics of IP3R with different concentrations of Ca2+ and IP3 should be more carefully addressed when new models for IP3R are developed.
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The application of nonlinear Dynamic Causal Modelling for fMRI in subjects at high genetic risk of schizophrenia. Neuroimage 2013; 73:16-29. [PMID: 23384525 DOI: 10.1016/j.neuroimage.2013.01.063] [Citation(s) in RCA: 41] [Impact Index Per Article: 3.7] [Reference Citation Analysis] [Abstract] [Journal Information] [Subscribe] [Scholar Register] [Received: 09/17/2012] [Revised: 01/17/2013] [Accepted: 01/22/2013] [Indexed: 01/22/2023] Open
Abstract
Nonlinear Dynamic Causal Modelling (DCM) for fMRI provides computational modelling of gating mechanisms at the neuronal population level. It allows for estimations of connection strengths with nonlinear modulation within task-dependent networks. This paper presents an application of nonlinear DCM in subjects at high familial risk of schizophrenia performing the Hayling Sentence Completion Task (HSCT). We analysed scans of 19 healthy controls and 46 subjects at high familial risk of schizophrenia, which included 26 high risk subjects without psychotic symptoms and 20 subjects with psychotic symptoms. The activity-dependent network consists of the intra parietal cortex (IPS), inferior frontal gyrus (IFG), middle temporal gyrus (MTG), anterior cingulate cortex (ACC) and the mediodorsal (MD) thalamus. The connections between the MD thalamus and the IFG were gated by the MD thalamus. We used DCM to investigate altered connection strength of these connections. Bayesian Model Selection (BMS) at the group and family level was used to compare the optimal bilinear and nonlinear models. Bayesian Model Averaging (BMA) was used to assess the connection strengths with the gating from the MD thalamus and the IFG. The nonlinear models provided the better explanation of the data. Furthermore, the BMA analysis showed significantly lower connection strength of the thalamocortical connection with nonlinear modulation from the MD thalamus in high risk subjects with psychotic symptoms and those who subsequently developed schizophrenia. These findings demonstrate that nonlinear DCM provides a method to investigate altered connectivity at the level of neural circuits. The reduced connection strength with thalamic gating may be a neurobiomarker implicated in the development of psychotic symptoms. This study suggests that nonlinear DCM could lead to new insights into functional and effective dysconnection at the network level in subjects at high familial risk.
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Güler M. Stochastic Hodgkin-Huxley Equations with Colored Noise Terms in the Conductances. Neural Comput 2013; 25:46-74. [DOI: 10.1162/neco_a_00384] [Citation(s) in RCA: 16] [Impact Index Per Article: 1.5] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/04/2022]
Abstract
The excitability of cells is facilitated by voltage-gated ion channels. These channels accommodate a multiple number of gates individually. The possible impact of that gate multiplicity on the cell's function, specifically when the membrane area is of limited size, was investigated in the author's prior work (Güler, 2011 ). There, it was found that a nontrivially persistent correlation takes place between the transmembrane voltage fluctuations (also between the fluctuations in the gating variables) and the component of open channel fluctuations attributed to the gate multiplicity. This nontrivial phenomenon was found to be playing a major augmentative role for the elevation of excitability and spontaneous firing in small cells. In addition, the same phenomenon was found to be enhancing spike coherence significantly. Here we extend Fox and Lu's ( 1994 ) stochastic Hodgkin-Huxley equations by incorporating colored noise terms into the conductances there to obtain a formalism capable of capturing the addressed cross-correlations. Statistics of spike generation, spike coherence, firing efficiency, latency, and jitter from the articulated set of equations are found to be highly accurate in comparison with the corresponding statistics from the exact microscopic Markov simulations. This way, it is demonstrated vividly that our formulation overcomes the inherent inadequacy of the Fox and Lu equations. Finally, a recently proposed diffusion approximation method (Linaro, Storace, & Giugliano, 2011 ) is taken into consideration, and a discussion on its character is pursued.
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Affiliation(s)
- Marifi Güler
- Department of Computer Engineering, Eastern Mediterranean University, Famagusta, via Mersin-10, Turkey
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Huang Y, Rüdiger S, Shuai J. Channel-based Langevin approach for the stochastic Hodgkin-Huxley neuron. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2013; 87:012716. [PMID: 23410368 DOI: 10.1103/physreve.87.012716] [Citation(s) in RCA: 8] [Impact Index Per Article: 0.7] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Subscribe] [Scholar Register] [Received: 09/17/2012] [Revised: 12/07/2012] [Indexed: 06/01/2023]
Abstract
Stochasticity in ion channel gating is the major source of intrinsic neuronal noise, which can induce many important effects in neuronal dynamics. Several numerical implementations of the Langevin approach have been proposed to approximate the Markovian dynamics of the Hodgkin-Huxley neuronal model. In this work an improved channel-based Langevin approach is proposed by introducing a truncation procedure to limit the state fractions in the range of [0, 1]. The truncated fractions are put back into the state fractions in the next time step for channel noise calculation. Our simulations show that the bounded Langevin approaches combined with the restored process give better approximations to the statistics of action potentials with the Markovian method. As a result, in our approach the channel state fractions are disturbed by two terms of noise: an uncorrelated Gaussian noise and a time-correlated noise obtained from the truncated fractions. We suggest that the restoration of truncated fractions is a critical process for a bounded Langevin method.
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Affiliation(s)
- Yandong Huang
- Department of Physics and Institute of Theoretical Physics and Astrophysics, Xiamen University, Xiamen 361005, China
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The effect of neural noise on spike time precision in a detailed CA3 neuron model. COMPUTATIONAL AND MATHEMATICAL METHODS IN MEDICINE 2012; 2012:595398. [PMID: 22778784 PMCID: PMC3388596 DOI: 10.1155/2012/595398] [Citation(s) in RCA: 5] [Impact Index Per Article: 0.4] [Reference Citation Analysis] [Abstract] [Track Full Text] [Download PDF] [Figures] [Subscribe] [Scholar Register] [Received: 09/30/2011] [Revised: 12/21/2011] [Accepted: 01/23/2012] [Indexed: 11/26/2022]
Abstract
Experimental and computational studies emphasize the role of the millisecond precision of neuronal spike times as an important coding mechanism for transmitting and representing information in the central nervous system. We investigate the spike time precision of a multicompartmental pyramidal neuron model of the CA3 region of the hippocampus under the influence of various sources of neuronal noise. We describe differences in the contribution to noise originating from voltage-gated ion channels, synaptic vesicle release, and vesicle quantal size. We analyze the effect of interspike intervals and the voltage course preceding the firing of spikes on the spike-timing jitter. The main finding of this study is the ranking of different noise sources according to their contribution to spike time precision. The most influential is synaptic vesicle release noise, causing the spike jitter to vary from 1 ms to 7 ms of a mean value 2.5 ms. Of second importance was the noise incurred by vesicle quantal size variation causing the spike time jitter to vary from 0.03 ms to 0.6 ms. Least influential was the voltage-gated channel noise generating spike jitter from 0.02 ms to 0.15 ms.
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Increased excitability and altered action potential waveform in cerebellar granule neurons of the Ts65Dn mouse model of Down syndrome. Brain Res 2012; 1465:10-7. [PMID: 22627164 PMCID: PMC3389345 DOI: 10.1016/j.brainres.2012.05.027] [Citation(s) in RCA: 12] [Impact Index Per Article: 1.0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 03/23/2012] [Revised: 05/09/2012] [Accepted: 05/14/2012] [Indexed: 12/19/2022]
Abstract
Down syndrome (DS) is characterized by intellectual disability and impaired motor control. Lack of coordinated movement, poor balance, and unclear speech imply dysfunction of the cerebellum, which is known to be reduced in volume in DS. The principal cause of the smaller cerebellum is a diminished number of granule cells (GCs). These neurons form the ‘input layer’ of the cerebellar cortex, where sensorimotor information carried by incoming mossy fibers is transformed before it is conveyed to Purkinje cells and inhibitory interneurons. However, it is not known how processing of this information is affected in the hypogranular cerebellum that characterizes DS. Here we explore the possibility that the electrical properties of the surviving GCs are changed. We find that in the Ts65Dn mouse model of DS, GCs have a higher input resistance at voltages approaching the threshold for firing, which causes them to be more excitable. In addition, they fire narrower and larger amplitude action potentials. These subtly modified electrical properties may result in atypical transfer of information at the input layer of the cerebellum.
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Abstract
Conductance-based equations for electrically active cells form one of the most widely studied mathematical frameworks in computational biology. This framework, as expressed through a set of differential equations by Hodgkin and Huxley, synthesizes the impact of ionic currents on a cell's voltage--and the highly nonlinear impact of that voltage back on the currents themselves--into the rapid push and pull of the action potential. Later studies confirmed that these cellular dynamics are orchestrated by individual ion channels, whose conformational changes regulate the conductance of each ionic current. Thus, kinetic equations familiar from physical chemistry are the natural setting for describing conductances; for small-to-moderate numbers of channels, these will predict fluctuations in conductances and stochasticity in the resulting action potentials. At first glance, the kinetic equations provide a far more complex (and higher-dimensional) description than the original Hodgkin-Huxley equations or their counterparts. This has prompted more than a decade of efforts to capture channel fluctuations with noise terms added to the equations of Hodgkin-Huxley type. Many of these approaches, while intuitively appealing, produce quantitative errors when compared to kinetic equations; others, as only very recently demonstrated, are both accurate and relatively simple. We review what works, what doesn't, and why, seeking to build a bridge to well-established results for the deterministic equations of Hodgkin-Huxley type as well as to more modern models of ion channel dynamics. As such, we hope that this review will speed emerging studies of how channel noise modulates electrophysiological dynamics and function. We supply user-friendly MATLAB simulation code of these stochastic versions of the Hodgkin-Huxley equations on the ModelDB website (accession number 138950) and http://www.amath.washington.edu/~etsb/tutorials.html.
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Affiliation(s)
- Joshua H Goldwyn
- Department of Applied Mathematics, University of Washington, Seattle, Washington, USA.
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Effects of transmitters and amyloid-beta peptide on calcium signals in rat cortical astrocytes: Fura-2AM measurements and stochastic model simulations. PLoS One 2011; 6:e17914. [PMID: 21483471 PMCID: PMC3066169 DOI: 10.1371/journal.pone.0017914] [Citation(s) in RCA: 34] [Impact Index Per Article: 2.6] [Reference Citation Analysis] [Abstract] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 12/09/2010] [Accepted: 02/14/2011] [Indexed: 12/03/2022] Open
Abstract
Background To better understand the complex molecular level interactions seen in the
pathogenesis of Alzheimer's disease, the results of the wet-lab and
clinical studies can be complemented by mathematical models. Astrocytes are
known to become reactive in Alzheimer's disease and their ionic
equilibrium can be disturbed by interaction of the released and accumulated
transmitters, such as serotonin, and peptides, including
amyloid- peptides
(A). We have here studied the effects of small amounts
of A25–35 fragments on the transmitter-induced
calcium signals in astrocytes by Fura-2AM fluorescence measurements and
running simulations of the detected calcium signals. Methodology/Principal Findings Intracellular calcium signals were measured in cultured rat cortical
astrocytes following additions of serotonin and glutamate, or either of
these transmitters together with A25–35.
A25–35 increased the number of astrocytes
responding to glutamate and exceedingly increased the magnitude of the
serotonin-induced calcium signals. In addition to
A25–35-induced effects, the contribution of
intracellular calcium stores to calcium signaling was tested. When using
higher stimulus frequency, the subsequent calcium peaks after the initial
peak were of lower amplitude. This may indicate inadequate filling of the
intracellular calcium stores between the stimuli. In order to reproduce the
experimental findings, a stochastic computational model was introduced. The
model takes into account the major mechanisms known to be involved in
calcium signaling in astrocytes. Model simulations confirm the principal
experimental findings and show the variability typical for experimental
measurements. Conclusions/Significance Nanomolar A25–35 alone does not cause persistent change in
the basal level of calcium in astrocytes. However, even small amounts of
A25–35, together with transmitters, can have
substantial synergistic effects on intracellular calcium signals.
Computational modeling further helps in understanding the mechanisms
associated with intracellular calcium oscillations. Modeling the mechanisms
is important, as astrocytes have an essential role in regulating the
neuronal microenvironment of the central nervous system.
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Accurate and fast simulation of channel noise in conductance-based model neurons by diffusion approximation. PLoS Comput Biol 2011; 7:e1001102. [PMID: 21423712 PMCID: PMC3053314 DOI: 10.1371/journal.pcbi.1001102] [Citation(s) in RCA: 66] [Impact Index Per Article: 5.1] [Reference Citation Analysis] [Abstract] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 09/16/2010] [Accepted: 01/28/2011] [Indexed: 11/19/2022] Open
Abstract
Stochastic channel gating is the major source of intrinsic neuronal noise whose functional consequences at the microcircuit- and network-levels have been only partly explored. A systematic study of this channel noise in large ensembles of biophysically detailed model neurons calls for the availability of fast numerical methods. In fact, exact techniques employ the microscopic simulation of the random opening and closing of individual ion channels, usually based on Markov models, whose computational loads are prohibitive for next generation massive computer models of the brain. In this work, we operatively define a procedure for translating any Markov model describing voltage- or ligand-gated membrane ion-conductances into an effective stochastic version, whose computer simulation is efficient, without compromising accuracy. Our approximation is based on an improved Langevin-like approach, which employs stochastic differential equations and no Montecarlo methods. As opposed to an earlier proposal recently debated in the literature, our approximation reproduces accurately the statistical properties of the exact microscopic simulations, under a variety of conditions, from spontaneous to evoked response features. In addition, our method is not restricted to the Hodgkin-Huxley sodium and potassium currents and is general for a variety of voltage- and ligand-gated ion currents. As a by-product, the analysis of the properties emerging in exact Markov schemes by standard probability calculus enables us for the first time to analytically identify the sources of inaccuracy of the previous proposal, while providing solid ground for its modification and improvement we present here. A possible approach to understanding the neuronal bases of the computational properties of the nervous system consists of modelling its basic building blocks, neurons and synapses, and then simulating their collective activity emerging in large networks. In developing such models, a satisfactory description level must be chosen as a compromise between simplicity and faithfulness in reproducing experimental data. Deterministic neuron models – i.e., models that upon repeated simulation with fixed parameter values provide the same results – are usually made up of ordinary differential equations and allow for relatively fast simulation times. By contrast, they do not describe accurately the underlying stochastic response properties arising from the microscopical correlate of neuronal excitability. Stochastic models are usually based on mathematical descriptions of individual ion channels, or on an effective macroscopic account of their random opening and closing. In this contribution we describe a general method to transform any deterministic neuron model into its effective stochastic version that accurately replicates the statistical properties of ion channels random kinetics.
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A mathematical model of adult GnRH neurons in mouse brain and its bifurcation analysis. J Theor Biol 2011; 276:22-34. [PMID: 21300070 DOI: 10.1016/j.jtbi.2011.01.035] [Citation(s) in RCA: 27] [Impact Index Per Article: 2.1] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 07/08/2010] [Revised: 01/24/2011] [Accepted: 01/24/2011] [Indexed: 11/21/2022]
Abstract
GnRH neurons are hypothalamic neurons that secrete gonadotropin-releasing hormone (GnRH) which stimulates the release of gonadotropins, one of the crucial hormones for sexual development, fertility and maturation. A mathematical model was built to help elucidate the mechanisms underlying electrical bursting and synchronous [Ca²(+)] transients in GnRH neurons (Lee et al., 2010). The model predicted that bursting in GnRH neurons (at least of the short-bursting type) requires the existence of a [Ca²(+)]-dependent slow after-hyperpolarisation current (sI(AHP-UCL)), and this predicted current was found experimentally. GnRH behaviour under a wide range of conditions (inhibition of Na(+) channels, IP₃ receptors, [Ca²(+)]-dependent K(+) channels, or Ca²(+) pumps, or in the presence of zero extracellular [Ca²(+)]) is successfully reproduced by the model. In this paper, a simplified version of the previous model, with the same qualitative behaviour, is constructed and studied using timescale separation techniques and bifurcation analysis.
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Cannon RC, O'Donnell C, Nolan MF. Stochastic ion channel gating in dendritic neurons: morphology dependence and probabilistic synaptic activation of dendritic spikes. PLoS Comput Biol 2010; 6. [PMID: 20711353 PMCID: PMC2920836 DOI: 10.1371/journal.pcbi.1000886] [Citation(s) in RCA: 78] [Impact Index Per Article: 5.6] [Reference Citation Analysis] [Abstract] [MESH Headings] [Grants] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 06/17/2009] [Accepted: 07/14/2010] [Indexed: 11/18/2022] Open
Abstract
Neuronal activity is mediated through changes in the probability of stochastic transitions between open and closed states of ion channels. While differences in morphology define neuronal cell types and may underlie neurological disorders, very little is known about influences of stochastic ion channel gating in neurons with complex morphology. We introduce and validate new computational tools that enable efficient generation and simulation of models containing stochastic ion channels distributed across dendritic and axonal membranes. Comparison of five morphologically distinct neuronal cell types reveals that when all simulated neurons contain identical densities of stochastic ion channels, the amplitude of stochastic membrane potential fluctuations differs between cell types and depends on sub-cellular location. For typical neurons, the amplitude of membrane potential fluctuations depends on channel kinetics as well as open probability. Using a detailed model of a hippocampal CA1 pyramidal neuron, we show that when intrinsic ion channels gate stochastically, the probability of initiation of dendritic or somatic spikes by dendritic synaptic input varies continuously between zero and one, whereas when ion channels gate deterministically, the probability is either zero or one. At physiological firing rates, stochastic gating of dendritic ion channels almost completely accounts for probabilistic somatic and dendritic spikes generated by the fully stochastic model. These results suggest that the consequences of stochastic ion channel gating differ globally between neuronal cell-types and locally between neuronal compartments. Whereas dendritic neurons are often assumed to behave deterministically, our simulations suggest that a direct consequence of stochastic gating of intrinsic ion channels is that spike output may instead be a probabilistic function of patterns of synaptic input to dendrites. The activity of neurons in the brain is mediated through changes in the probability of random transitions between open and closed states of ion channels. Since differences in morphology define distinct types of neuron and may underlie neurological disorders, it is important to understand how morphology influences the functional consequences of these random transitions. However, the complexities of neuronal morphology, together with the large number of ion channels expressed by a single neuron, have made this issue difficult to explore systematically. We introduce and validate new computational tools that enable efficient generation and simulation of models containing ion channels distributed across complex neuronal morphologies. Using these tools we demonstrate that the impact of random ion channel opening depends on neuronal morphology and ion channel kinetics. We show that in a realistic model of a neuron important for navigation and memory random gating of ion channels substantially modifies responses to synaptic input. Our results suggest a new and general perspective, whereby output from a neuron is a probabilistic rather than a fixed function of synaptic input to its dendrites. These results and new tools will contribute to the understanding of how intrinsic properties of neurons influence computations carried out within the brain.
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Affiliation(s)
| | - Cian O'Donnell
- Neuroinformatics Doctoral Training Centre, University of Edinburgh, Edinburgh, United Kingdom
| | - Matthew F. Nolan
- Centre for Integrative Physiology, University of Edinburgh, Edinburgh, United Kingdom
- * E-mail:
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Tuckwell HC, Jost J. Weak noise in neurons may powerfully inhibit the generation of repetitive spiking but not its propagation. PLoS Comput Biol 2010; 6:e1000794. [PMID: 20523741 PMCID: PMC2877724 DOI: 10.1371/journal.pcbi.1000794] [Citation(s) in RCA: 23] [Impact Index Per Article: 1.6] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 12/15/2009] [Accepted: 04/23/2010] [Indexed: 11/24/2022] Open
Abstract
Many neurons have epochs in which they fire action potentials in an approximately periodic fashion. To see what effects noise of relatively small amplitude has on such repetitive activity we recently examined the response of the Hodgkin-Huxley (HH) space-clamped system to such noise as the mean and variance of the applied current vary, near the bifurcation to periodic firing. This article is concerned with a more realistic neuron model which includes spatial extent. Employing the Hodgkin-Huxley partial differential equation system, the deterministic component of the input current is restricted to a small segment whereas the stochastic component extends over a region which may or may not overlap the deterministic component. For mean values below, near and above the critical values for repetitive spiking, the effects of weak noise of increasing strength is ascertained by simulation. As in the point model, small amplitude noise near the critical value dampens the spiking activity and leads to a minimum as noise level increases. This was the case for both additive noise and conductance-based noise. Uniform noise along the whole neuron is only marginally more effective in silencing the cell than noise which occurs near the region of excitation. In fact it is found that if signal and noise overlap in spatial extent, then weak noise may inhibit spiking. If, however, signal and noise are applied on disjoint intervals, then the noise has no effect on the spiking activity, no matter how large its region of application, though the trajectories are naturally altered slightly by noise. Such effects could not be discerned in a point model and are important for real neuron behavior. Interference with the spike train does nevertheless occur when the noise amplitude is larger, even when noise and signal do not overlap, being due to the instigation of secondary noise-induced wave phenomena rather than switching the system from one attractor (firing regularly) to another (a stable point).
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Affiliation(s)
- Henry C Tuckwell
- Max Planck Institute for Mathematics in the Sciences, Leipzig, Germany.
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Kononenko NI, Berezetskaya NM. Modeling the spontaneous activity in suprachiasmatic nucleus neurons: role of cation single channels. J Theor Biol 2010; 265:115-25. [PMID: 20362589 DOI: 10.1016/j.jtbi.2010.03.039] [Citation(s) in RCA: 4] [Impact Index Per Article: 0.3] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 07/01/2009] [Revised: 03/26/2010] [Accepted: 03/28/2010] [Indexed: 11/29/2022]
Abstract
A population of interconnected neurons of the mammalian suprachiasmatic nuclei (SCN) controls circadian rhythms in physiological functions. In turn, a circadian rhythm of individual neurons is driven by intracellular processes, which via activation of specific membrane channels, produce circadian modulation of electrical firing rate. Yet the membrane target(s) of the cellular clock have remained enigmatic. Previously, subthreshold voltage-dependent cation (SVC) channels have been proposed as the membrane target of the cellular clock responsible for circadian modulation of the firing rate in SCN neurons. We tested this hypothesis with computational modeling based on experimental results from on-cell recording of SVC channel openings in acutely isolated SCN neurons and long-term continuous recording of activity from dispersed SCN neurons in a multielectrode array dish (MED). The model reproduced the circadian behavior if the number of SVC channels or their kinetics were modulated in accordance with protein concentration in a model of the intracellular clock (Scheper et al., 1999. J. Neurosci. 19, 40-47). Such modulation changed the average firing rate of the model neuron from zero ("subjective-night" silence) up to 18 Hz ("subjective-day" peak). Furthermore, the variability of interspike intervals (ISI) and the circadian pattern of firing rate (i.e. silence-to-activity ratio and shape of circadian peaks) are in reasonable agreement with experimental data obtained in dispersed SCN neurons in MED. These results suggest that the variability of ISI in intact SCN neurons is mostly due to stochastic single-channel openings, and that the circadian pattern of the firing rate is specified by threshold properties of dependence of the spontaneous firing rate on the number of single channels (R-N relationship). This plausible mathematical modeling supports the hypothesis that SVC channels could be a critical element in circadian modulation of firing rate in SCN neurons.
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Affiliation(s)
- Nikolai I Kononenko
- Department of General Physiology of Nervous System, Institute of Physiology, 4, Bogomoletz street, Kiev 01024, Ukraine.
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Daunizeau J, Friston K, Kiebel S. Variational Bayesian identification and prediction of stochastic nonlinear dynamic causal models. PHYSICA D. NONLINEAR PHENOMENA 2009; 238:2089-2118. [PMID: 19862351 PMCID: PMC2767160 DOI: 10.1016/j.physd.2009.08.002] [Citation(s) in RCA: 116] [Impact Index Per Article: 7.7] [Reference Citation Analysis] [Abstract] [Key Words] [Grants] [Track Full Text] [Figures] [Subscribe] [Scholar Register] [Received: 07/02/2008] [Revised: 07/29/2009] [Accepted: 08/01/2009] [Indexed: 05/28/2023]
Abstract
In this paper, we describe a general variational Bayesian approach for approximate inference on nonlinear stochastic dynamic models. This scheme extends established approximate inference on hidden-states to cover: (i) nonlinear evolution and observation functions, (ii) unknown parameters and (precision) hyperparameters and (iii) model comparison and prediction under uncertainty. Model identification or inversion entails the estimation of the marginal likelihood or evidence of a model. This difficult integration problem can be finessed by optimising a free-energy bound on the evidence using results from variational calculus. This yields a deterministic update scheme that optimises an approximation to the posterior density on the unknown model variables. We derive such a variational Bayesian scheme in the context of nonlinear stochastic dynamic hierarchical models, for both model identification and time-series prediction. The computational complexity of the scheme is comparable to that of an extended Kalman filter, which is critical when inverting high dimensional models or long time-series. Using Monte-Carlo simulations, we assess the estimation efficiency of this variational Bayesian approach using three stochastic variants of chaotic dynamic systems. We also demonstrate the model comparison capabilities of the method, its self-consistency and its predictive power.
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Affiliation(s)
- J. Daunizeau
- Corresponding address: Wellcome Trust for Neuroimaging, Institute of Neurology, UCL, 12 Queen Square, London, WC1N 3BG, United Kingdom. Tel.: +44 207 833 7488; fax: +44 207 813 1445.
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