1
|
Morrison CL, Greenwood PE, Ward LM. Plastic systemic inhibition controls amplitude while allowing phase pattern in a stochastic neural field model. Phys Rev E 2021; 103:032311. [PMID: 33862754 DOI: 10.1103/physreve.103.032311] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.3] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 10/09/2020] [Accepted: 02/19/2021] [Indexed: 11/07/2022]
Abstract
We investigate oscillatory phase pattern formation and amplitude control for a linearized stochastic neuron field model by simulating Mexican-hat-coupled stochastic processes. We find, for several choices of parameters, that spatial pattern formation in the temporal phases of the coupled processes occurs if and only if their amplitudes are allowed to grow unrealistically large. Stimulated by recent work on homeostatic inhibitory plasticity, we introduce static and plastic (adaptive) systemic inhibitory mechanisms to keep the amplitudes stochastically bounded. We find that systems with static inhibition exhibited bounded amplitudes but no sustained phase patterns. With plastic systemic inhibition, on the other hand, the resulting systems exhibit both bounded amplitudes and sustained phase patterns. These results demonstrate that plastic inhibitory mechanisms in neural field models can dynamically control amplitudes while allowing patterns of phase synchronization to develop. Similar mechanisms of plastic systemic inhibition could play a role in regulating oscillatory functioning in the brain.
Collapse
Affiliation(s)
- Conor L Morrison
- Department of Statistics, University of British Columbia, Vancouver, British Columbia, Canada V6T 1Z4
| | - Priscilla E Greenwood
- Department of Mathematics, University of British Columbia, Vancouver, British Columbia, Canada V6T 1Z2
| | - Lawrence M Ward
- Department of Psychology and Djavad Mowafaghian Centre for Brain Health, 2136 West Mall, University of British Columbia, Vancouver, British Columbia, Canada V6T 1Z4
| |
Collapse
|
2
|
Byrne Á, Avitabile D, Coombes S. Next-generation neural field model: The evolution of synchrony within patterns and waves. Phys Rev E 2019; 99:012313. [PMID: 30780315 DOI: 10.1103/physreve.99.012313] [Citation(s) in RCA: 11] [Impact Index Per Article: 2.2] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 09/07/2018] [Indexed: 05/01/2023]
Abstract
Neural field models are commonly used to describe wave propagation and bump attractors at a tissue level in the brain. Although motivated by biology, these models are phenomenological in nature. They are built on the assumption that the neural tissue operates in a near synchronous regime, and hence, cannot account for changes in the underlying synchrony of patterns. It is customary to use spiking neural network models when examining within population synchronization. Unfortunately, these high-dimensional models are notoriously hard to obtain insight from. In this paper, we consider a network of θ-neurons, which has recently been shown to admit an exact mean-field description in the absence of a spatial component. We show that the inclusion of space and a realistic synapse model leads to a reduced model that has many of the features of a standard neural field model coupled to a further dynamical equation that describes the evolution of network synchrony. Both Turing instability analysis and numerical continuation software are used to explore the existence and stability of spatiotemporal patterns in the system. In particular, we show that this new model can support states above and beyond those seen in a standard neural field model. These states are typified by structures within bumps and waves showing the dynamic evolution of population synchrony.
Collapse
Affiliation(s)
- Áine Byrne
- Center for Neural Science, New York University, New York, New York 10003, USA and Centre for Mathematical Medicine and Biology, School of Mathematical Sciences, University of Nottingham, University Park, Nottingham, NG7 2RD, United Kingdom
| | - Daniele Avitabile
- Centre for Mathematical Medicine and Biology, School of Mathematical Sciences, University of Nottingham, University Park, Nottingham, NG7 2RD, United Kingdom and Inria Sophia Antipolis Méditerranée Research Centre, MathNeuro Team, 2004 route des Lucioles, Bote Postale 93 06902 Sophia Antipolis, Cedex, France
| | - Stephen Coombes
- Centre for Mathematical Medicine and Biology, School of Mathematical Sciences, University of Nottingham, University Park, Nottingham, NG7 2RD, United Kingdom
| |
Collapse
|
3
|
Romeo A, Supèr H. 'Two vs one' rivalry by the Loxley-Robinson model. BIOLOGICAL CYBERNETICS 2017; 111:405-420. [PMID: 28929217 DOI: 10.1007/s00422-017-0734-x] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Key Words] [MESH Headings] [Track Full Text] [Subscribe] [Scholar Register] [Received: 02/10/2017] [Accepted: 09/06/2017] [Indexed: 06/07/2023]
Abstract
We apply the competitive model of Loxley and Robinson (Phys Rev Lett 102:258701, 2009. doi: 10.1103/PhysRevLett.102.258701 ) to the study of a special case of visual rivalry. Three-peaked inputs with maxima at symmetrical locations are introduced, and the role of three-bump configurations is then considered. The model yields conditions for what can be interpreted as a bistable percept analogous to the one-dimensional version of a competition between the central and flanking parts of an image.
Collapse
Affiliation(s)
- August Romeo
- Department of Cognition, Development and Educational Psychology, Faculty of Psychology, University of Barcelona (UB), Barcelona, Spain
| | - Hans Supèr
- Department of Cognition, Development and Educational Psychology, Faculty of Psychology, University of Barcelona (UB), Barcelona, Spain.
- Institut de Neurociències, University of Barcelona (UB), Barcelona, Spain.
- Catalan Institution for Research and Advanced Studies (ICREA), Barcelona, Spain.
| |
Collapse
|
4
|
Gökçe A, Avitabile D, Coombes S. The Dynamics of Neural Fields on Bounded Domains: An Interface Approach for Dirichlet Boundary Conditions. JOURNAL OF MATHEMATICAL NEUROSCIENCE 2017; 7:12. [PMID: 29075933 PMCID: PMC5658324 DOI: 10.1186/s13408-017-0054-4] [Citation(s) in RCA: 3] [Impact Index Per Article: 0.4] [Reference Citation Analysis] [Abstract] [Key Words] [Grants] [Track Full Text] [Figures] [Subscribe] [Scholar Register] [Received: 03/20/2017] [Accepted: 10/10/2017] [Indexed: 06/07/2023]
Abstract
Continuum neural field equations model the large-scale spatio-temporal dynamics of interacting neurons on a cortical surface. They have been extensively studied, both analytically and numerically, on bounded as well as unbounded domains. Neural field models do not require the specification of boundary conditions. Relatively little attention has been paid to the imposition of neural activity on the boundary, or to its role in inducing patterned states. Here we redress this imbalance by studying neural field models of Amari type (posed on one- and two-dimensional bounded domains) with Dirichlet boundary conditions. The Amari model has a Heaviside nonlinearity that allows for a description of localised solutions of the neural field with an interface dynamics. We show how to generalise this reduced but exact description by deriving a normal velocity rule for an interface that encapsulates boundary effects. The linear stability analysis of localised states in the interface dynamics is used to understand how spatially extended patterns may develop in the absence and presence of boundary conditions. Theoretical results for pattern formation are shown to be in excellent agreement with simulations of the full neural field model. Furthermore, a numerical scheme for the interface dynamics is introduced and used to probe the way in which a Dirichlet boundary condition can limit the growth of labyrinthine structures.
Collapse
Affiliation(s)
- Aytül Gökçe
- School of Mathematical Sciences, University of Nottingham, University Park, Nottingham, NG7 2RD UK
| | - Daniele Avitabile
- School of Mathematical Sciences, University of Nottingham, University Park, Nottingham, NG7 2RD UK
| | - Stephen Coombes
- School of Mathematical Sciences, University of Nottingham, University Park, Nottingham, NG7 2RD UK
| |
Collapse
|
5
|
Romeo A, Supèr H. Bump competition and lattice solutions in two-dimensional neural fields. Neural Netw 2017; 94:141-158. [PMID: 28779599 DOI: 10.1016/j.neunet.2017.07.003] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.1] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 09/28/2016] [Revised: 05/19/2017] [Accepted: 07/02/2017] [Indexed: 10/19/2022]
Abstract
Some forms of competition among activity bumps in a two-dimensional neural field are studied. First, threshold dynamics is included and rivalry evolutions are considered. The relations between parameters and dominance durations can match experimental observations about ageing. Next, the threshold dynamics is omitted from the model and we focus on the properties of the steady-state. From noisy inputs, hexagonal grids are formed by a symmetry-breaking process. Particular issues about solution existence and stability conditions are considered. We speculate that they affect the possibility of producing basis grids which may be combined to form feature maps.
Collapse
Affiliation(s)
- August Romeo
- Department of Cognition, Development and Educational Psychology, Faculty of Psychology, University of Barcelona, Spain
| | - Hans Supèr
- Department of Cognition, Development and Educational Psychology, Faculty of Psychology, University of Barcelona, Spain; Institut de Neurociències, University of Barcelona, Spain; Catalan Institution for Research and Advanced Studies (ICREA), Spain.
| |
Collapse
|
6
|
Visser S, Nicks R, Faugeras O, Coombes S. Standing and travelling waves in a spherical brain model: The Nunez model revisited. PHYSICA D. NONLINEAR PHENOMENA 2017; 349:27-45. [PMID: 28626276 PMCID: PMC5421190 DOI: 10.1016/j.physd.2017.02.017] [Citation(s) in RCA: 17] [Impact Index Per Article: 2.4] [Reference Citation Analysis] [Abstract] [Key Words] [Grants] [Track Full Text] [Subscribe] [Scholar Register] [Received: 12/09/2016] [Revised: 02/27/2017] [Accepted: 02/28/2017] [Indexed: 06/07/2023]
Abstract
The Nunez model for the generation of electroencephalogram (EEG) signals is naturally described as a neural field model on a sphere with space-dependent delays. For simplicity, dynamical realisations of this model either as a damped wave equation or an integro-differential equation, have typically been studied in idealised one dimensional or planar settings. Here we revisit the original Nunez model to specifically address the role of spherical topology on spatio-temporal pattern generation. We do this using a mixture of Turing instability analysis, symmetric bifurcation theory, centre manifold reduction and direct simulations with a bespoke numerical scheme. In particular we examine standing and travelling wave solutions using normal form computation of primary and secondary bifurcations from a steady state. Interestingly, we observe spatio-temporal patterns which have counterparts seen in the EEG patterns of both epileptic and schizophrenic brain conditions.
Collapse
Affiliation(s)
- S. Visser
- School of Mathematical Sciences, University of Nottingham, NG7 2RD, UK
- Wellcome Trust Centre for Biomedical Modelling and Analysis, RILD Building, University of Exeter, EX2 5DW, UK
| | - R. Nicks
- School of Mathematical Sciences, University of Nottingham, NG7 2RD, UK
| | - O. Faugeras
- INRIA Sophia Antipolis Mediterannee, 2004 Route Des Lucioles, Sophia Antipolis, 06410, France
| | - S. Coombes
- School of Mathematical Sciences, University of Nottingham, NG7 2RD, UK
| |
Collapse
|
7
|
Thul R, Coombes S, Laing CR. Neural Field Models with Threshold Noise. JOURNAL OF MATHEMATICAL NEUROSCIENCE 2016; 6:3. [PMID: 26936267 PMCID: PMC4775726 DOI: 10.1186/s13408-016-0035-z] [Citation(s) in RCA: 2] [Impact Index Per Article: 0.3] [Reference Citation Analysis] [Abstract] [Key Words] [Grants] [Track Full Text] [Figures] [Subscribe] [Scholar Register] [Received: 11/26/2015] [Accepted: 02/19/2016] [Indexed: 06/05/2023]
Abstract
The original neural field model of Wilson and Cowan is often interpreted as the averaged behaviour of a network of switch like neural elements with a distribution of switch thresholds, giving rise to the classic sigmoidal population firing-rate function so prevalent in large scale neuronal modelling. In this paper we explore the effects of such threshold noise without recourse to averaging and show that spatial correlations can have a strong effect on the behaviour of waves and patterns in continuum models. Moreover, for a prescribed spatial covariance function we explore the differences in behaviour that can emerge when the underlying stationary distribution is changed from Gaussian to non-Gaussian. For travelling front solutions, in a system with exponentially decaying spatial interactions, we make use of an interface approach to calculate the instantaneous wave speed analytically as a series expansion in the noise strength. From this we find that, for weak noise, the spatially averaged speed depends only on the choice of covariance function and not on the shape of the stationary distribution. For a system with a Mexican-hat spatial connectivity we further find that noise can induce localised bump solutions, and using an interface stability argument show that there can be multiple stable solution branches.
Collapse
Affiliation(s)
- Rüdiger Thul
- Centre for Mathematical Medicine and Biology, School of Mathematical Sciences, University of Nottingham, University Park, Nottingham, NG7 2RD, UK.
| | - Stephen Coombes
- Centre for Mathematical Medicine and Biology, School of Mathematical Sciences, University of Nottingham, University Park, Nottingham, NG7 2RD, UK.
| | - Carlo R Laing
- Institute of Natural and Mathematical Sciences, Massey University (Albany), Private Bag 102-904, North Shore Mail Centre, Auckland, New Zealand.
| |
Collapse
|
8
|
Fard FS, Hollensen P, Heinke D, Trappenberg TP. Modeling human target reaching with an adaptive observer implemented with dynamic neural fields. Neural Netw 2015; 72:13-30. [PMID: 26559472 DOI: 10.1016/j.neunet.2015.10.003] [Citation(s) in RCA: 8] [Impact Index Per Article: 0.9] [Reference Citation Analysis] [Abstract] [Key Words] [MESH Headings] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 02/13/2015] [Revised: 09/30/2015] [Accepted: 10/06/2015] [Indexed: 11/18/2022]
Abstract
Humans can point fairly accurately to memorized states when closing their eyes despite slow or even missing sensory feedback. It is also common that the arm dynamics changes during development or from injuries. We propose a biologically motivated implementation of an arm controller that includes an adaptive observer. Our implementation is based on the neural field framework, and we show how a path integration mechanism can be trained from few examples. Our results illustrate successful generalization of path integration with a dynamic neural field by which the robotic arm can move in arbitrary directions and velocities. Also, by adapting the strength of the motor effect the observer implicitly learns to compensate an image acquisition delay in the sensory system. Our dynamic implementation of an observer successfully guides the arm toward the target in the dark, and the model produces movements with a bell-shaped velocity profile, consistent with human behavior data.
Collapse
Affiliation(s)
- Farzaneh S Fard
- Faculty of Computer Science, Dalhousie University, NS, Canada.
| | - Paul Hollensen
- Faculty of Computer Science, Dalhousie University, NS, Canada.
| | - Dietmar Heinke
- Center for Computational Neuroscience and Cognitive Robotics, School of Psychology, University of Birmingham, UK.
| | | |
Collapse
|
9
|
Qi Y, Gong P. Dynamic patterns in a two-dimensional neural field with refractoriness. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2015; 92:022702. [PMID: 26382427 DOI: 10.1103/physreve.92.022702] [Citation(s) in RCA: 8] [Impact Index Per Article: 0.9] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Subscribe] [Scholar Register] [Received: 03/30/2015] [Indexed: 06/05/2023]
Abstract
The formation of dynamic patterns such as localized propagating waves is a fascinating self-organizing phenomenon that happens in a wide range of spatially extended systems including neural systems, in which they might play important functional roles. Here we derive a type of two-dimensional neural-field model with refractoriness to study the formation mechanism of localized waves. After comparing this model with existing neural-field models, we show that it is able to generate a variety of localized patterns, including stationary bumps, localized waves rotating along a circular path, and localized waves with longer-range propagation. We construct explicit bump solutions for the two-dimensional neural field and conduct a linear stability analysis on how a stationary bump transitions to a propagating wave under different spatial eigenmode perturbations. The neural-field model is then partially solved in a comoving frame to obtain localized wave solutions, whose spatial profiles are in good agreement with those obtained from simulations. We demonstrate that when there are multiple such propagating waves, they exhibit rich propagation dynamics, including propagation along periodically oscillating and irregular trajectories; these propagation dynamics are quantitatively characterized. In addition, we show that these waves can have repulsive or merging collisions, depending on their collision angles and the refractoriness parameter. Due to its analytical tractability, the two-dimensional neural-field model provides a modeling framework for studying localized propagating waves and their interactions.
Collapse
Affiliation(s)
- Yang Qi
- School of Physics, University of Sydney, New South Wales 2006, Australia
| | - Pulin Gong
- School of Physics, University of Sydney, New South Wales 2006, Australia
| |
Collapse
|
10
|
Fotouhi M, Heidari M, Sharifitabar M. Continuous neural network with windowed Hebbian learning. BIOLOGICAL CYBERNETICS 2015; 109:321-332. [PMID: 25677526 DOI: 10.1007/s00422-015-0645-7] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.1] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Subscribe] [Scholar Register] [Received: 11/25/2013] [Accepted: 01/27/2015] [Indexed: 06/04/2023]
Abstract
We introduce an extension of the classical neural field equation where the dynamics of the synaptic kernel satisfies the standard Hebbian type of learning (synaptic plasticity). Here, a continuous network in which changes in the weight kernel occurs in a specified time window is considered. A novelty of this model is that it admits synaptic weight decrease as well as the usual weight increase resulting from correlated activity. The resulting equation leads to a delay-type rate model for which the existence and stability of solutions such as the rest state, bumps, and traveling fronts are investigated. Some relations between the length of the time window and the bump width is derived. In addition, the effect of the delay parameter on the stability of solutions is shown. Also numerical simulations for solutions and their stability are presented.
Collapse
Affiliation(s)
- M Fotouhi
- Department of Mathematical Sciences, Sharif University of Technology, P.O. Box 11365-9415, Tehran, Iran,
| | | | | |
Collapse
|