Neural ‘bubble’ dynamics in two dimensions: foundations

An extension to two dimensions of recent results in continuum neural field theory (CNFT) in one dimension is presented here. Focus is placed on the treatment of receptive fields and of learning on afferent synapses to obtain topographic maps. Received: 26 March 1997 / Accepted in revised form: 16 December 1998     

Localized states in an unbounded neural field equation with smooth firing rate function: a multi-parameter analysis

The existence of spatially localized solutions in neural networks is an important topic in neuroscience as these solutions are considered to characterize working (short-term) memory. We work with an unbounded neural network represented by the neural field equation with smooth firing rate function and a wizard hat spatial connectivity. Noting that stationary solutions of our neural field equation are equivalent to homoclinic orbits in a related fourth order ordinary differential equation, we apply normal form theory for a reversible Hopf bifurcation to prove the existence of localized solutions; further, we present results concerning their stability. Numerical continuation is used to compute branches of localized solution that exhibit snaking-type behaviour. We describe in terms of three parameters the exact regions for which localized solutions persist.     

Existence and stability of local excitations in homogeneous neural fields

Summary Dynamics of excitation patterns is studied in one-dimensional homogeneous lateral-inhibition type neural fields. The existence of a local excitation pattern solution as well as its waveform stability is proved by the use of the Schauder fixed-point theorem and a generalized version of the Perron-Frobenius theorem of positive matrices to the function space. The dynamics of the field is in general multi-stable so that the field can keep short-term memory.     

Spatial cognition and neuro-mimetic navigation: a model of hippocampal place cell activity

 A computational model of hippocampal activity during spatial cognition and navigation tasks is presented. The spatial representation in our model of the rat hippocampus is built on-line during exploration via two processing streams. An allothetic vision-based representation is built by unsupervised Hebbian learning extracting spatio-temporal properties of the environment from visual input. An idiothetic representation is learned based on internal movement-related information provided by path integration. On the level of the hippocampus, allothetic and idiothetic representations are integrated to yield a stable representation of the environment by a population of localized overlapping CA3-CA1 place fields. The hippocampal spatial representation is used as a basis for goal-oriented spatial behavior. We focus on the neural pathway connecting the hippocampus to the nucleus accumbens. Place cells drive a population of locomotor action neurons in the nucleus accumbens. Reward-based learning is applied to map place cell activity into action cell activity. The ensemble action cell activity provides navigational maps to support spatial behavior. We present experimental results obtained with a mobile Khepera robot. Received: 02 July 1999 / Accepted in revised form: 20 March 2000     

Periodic solutions in a model of competition between plasmid-bearing and plasmid-free organisms in a chemostat with an inhibitor

We obtain necessary and sufficient conditions on the existence of a unique positive equilibrium point and a set of sufficient conditions on the existence of periodic solutions for a 3-dimensional system which arises from a model of competition between plasmid-bearing and plasmid-free organisms in a chemostat with an inhibitor. Our results improve the corresponding results obtained by Hsu, Luo, and Waltman [1]. Received: 20 November 1997 / Revised version: 12 February 1999 / Published online: 20 December 2000     

The effect of dispersal on single-species nonautonomous dispersal models with delays

In this paper, single-species nonautonomous dispersal models with delays are considered. An interesting result on the effect of dispersal for persistence and extinction is obtained. That is, if the species is persistent in a patch then it is also persistent in all other patches; if the species is permanent in a patch then it is also permanent in all other patches; if the species is extinct in a patch then it is also extinct in all other patches. Furthermore, some new sufficient conditions for the permanence and extinction of the species in a patch are established. The existence of positive periodic solutions is obtained in the periodic case by employing Teng and Chen's results on the existence of positive periodic solutions for functional differential equations. Received: 26 June 2000 / Revised version: 6 October 2000 / Published online: 10 April 2001     

Tuned solutions in dynamic neural fields as building blocks for extended EEG models

The most prominent functional property of cortical neurons in sensory areas are their tuned receptive fields which provide specific responses of the neurons to external stimuli. Tuned neural firing indeed reflects the most basic and best worked out level of cognitive representations. Tuning properties can be dynamic on a short time-scale of fractions of a second. Such dynamic effects have been modeled by localised solutions (also called “bumps” or “peaks”) in dynamic neural fields. In the present work we develop an approximation method to reduce the dynamics of localised activation peaks in systems of n coupled nonlinear d-dimensional neural fields with transmission delays to a small set of delay differential equations for the peak amplitudes and widths only. The method considerably simplifies the analysis of peaked solutions as demonstrated for a two-dimensional example model of neural feature selectivity in the brain. The reduced equations describe the effective interaction between pools of local neurons of several (n) classes that participate in shaping the dynamic receptive field responses. To lowest order they resemble neural mass models as they often form the base of EEG-models. Thereby they provide a link between functional small-scale receptive field models and more coarse-grained EEG-models. More specifically, they connect the dynamics in feature-selective cortical microcircuits to the more abstract local elements used in coarse-grained models. However, beside amplitudes the reduced equations also reflect the sharpness of tuning of the activity in a d-dimensional feature space in response to localised stimuli.     

Travelling wave phenomena in non-linear diffusion degenerate Nagumo equations

 In this paper we study the existence of one-dimensional travelling wave solutions u(x, t)=φ(x−ct) for the non-linear degenerate (at u=0) reaction-diffusion equation u _t =[D(u)u _x ] _x +g(u) where g is a generalisation of the Nagumo equation arising in nerve conduction theory, as well as describing the Allee effect. We use a dynamical systems approach to prove: 1. the global bifurcation of a heteroclinic cycle (two monotone stationary front solutions), for c=0, 2. The existence of a unique value c ^*>0 of c for which φ(x−c ^* t) is a travelling wave solution of sharp type and 3. A continuum of monotone and oscillatory fronts for c≠c ^*. We present some numerical simulations of the phase portrait in travelling wave coordinates and on the full partial differential equation. Received 15 December 1995; received in revised form 14 May 1996     

Traveling waves and pulses in a one-dimensional network of excitable integrate-and-fire neurons

 We study the existence and stability of traveling waves and pulses in a one-dimensional network of integrate-and-fire neurons with synaptic coupling. This provides a simple model of excitable neural tissue. We first derive a self-consistency condition for the existence of traveling waves, which generates a dispersion relation between velocity and wavelength. We use this to investigate how wave-propagation depends on various parameters that characterize neuronal interactions such as synaptic and axonal delays, and the passive membrane properties of dendritic cables. We also establish that excitable networks support the propagation of solitary pulses in the long-wavelength limit. We then derive a general condition for the (local) asymptotic stability of traveling waves in terms of the characteristic equation of the linearized firing time map, which takes the form of an integro-difference equation of infinite order. We use this to analyze the stability of solitary pulses in the long-wavelength limit. Solitary wave solutions are shown to come in pairs with the faster (slower) solution stable (unstable) in the case of zero axonal delays; for non-zero delays and fast synapses the stable wave can itself destabilize via a Hopf bifurcation. Received: 27 October 1998     

Wave propagation in discrete media

 We have considered infinite systems of nonlinear ODEs on the one-dimensional integer lattice which describes the activity in an excitatorily coupled network of excitable cells. For an ideal nonlinearity, we calculated the speed of propagation of an activity and derived the condition for its existence. We also studied the existence and stability of the traveling wave solution and gave, in the simplest case, its explicit expression. We established that some unstable traveling waves lead to propagation with an enlarging profile defined by a front velocity and a wake velocity. We generalized some results to inhomogeneous medium and network with long range connections. Received: 3 July 2000 / Revised version: 17 April 2001 / Published online: 7 December 2001     

Existence of traveling wave solutions in a diffusive predator-prey model

 We establish the existence of traveling front solutions and small amplitude traveling wave train solutions for a reaction-diffusion system based on a predator-prey model with Holling type-II functional response. The traveling front solutions are equivalent to heteroclinic orbits in R ⁴ and the small amplitude traveling wave train solutions are equivalent to small amplitude periodic orbits in R ⁴ . The methods used to prove the results are the shooting argument and the Hopf bifurcation theorem. Received: 25 May 2001 / Revised version: 5 August 2002 / Published online: 19 November 2002 RID="*" ID="*" Research was supported by the National Natural Science Foundations (NNSF) of China. RID="*" ID="*" Research was partially supported by the Natural Sciences and Engineering Research Council (NSERC) of Canada. On leave from the Department of Mathematics and Statistics, Dalhousie University, Halifax, Nova Scotia B3H 3J5, Canada. Mathematics Subject Classification (2000): 34C35, 35K57 Key words or phrases: Traveling wave solution – Wazewski set – Shooting argument – Hopf bifurcation Acknowledgements. We would like to thank the two referees for their careful reading and helpful comments.     

Dynamics of pattern formation in lateral-inhibition type neural fields

The dynamics of pattern formation is studied for lateral-inhibition type homogeneous neural fields with general connections. Neural fields consisting of single layer are first treated, and it is proved that there are five types of pattern dynamics. The type of the dynamics of a field depends not only on the mutual connections within the field but on the level of homogeneous stimulus given to the field. An example of the dynamics is as follows: A fixed size of localized excitation, once evoked by stimulation, can be retained in the field persistently even after the stimulation vanishes. It moves until it finds the position of the maximum of the input stimulus. Fields consisting of an excitatory and an inhibitory layer are next analyzed. In addition to stationary localized excitation, fields have such pattern dynamics as production of oscillatory waves, travelling waves, active and dual active transients, etc.This research was supported in part by a Sloan Foundation grant to the Center for Systems Neuroscience, University of Massachusetts at AmherstOn sabbatical leave from the University of Tokyo     

Drift modes of a quasi-two-dimensional current sheet

Stability of a plasma configuration consisting of a thin one-dimensional current sheet embedded into a two-dimensional background current sheet is studied. Drift modes developing in plasma as unstable waves along the current direction are considered. Dispersion relations for kink and sausage perturbation modes are obtained depending on the ratio of parameters of thin and background current sheets. It is shown that the existence of the background sheet results in a decrease in the instability growth rates and a significant increase in the perturbation wavelengths. The role of drift modes in the excitation of oscillations observed in the current sheet of the Earth’s magnetotail is discussed.     

Characterization,scaling, and partial representation of neural junctions and coordinated firing patterns by dynamic similarity

 This paper presents a dynamic-similarity-based system for mathematically characterizing the functional connectivity and information flow of neural junctions. This approach allows for quantitative comparison of operations of neural junctions across systems, and an interpretation of their connectivity parameters in terms of the flow of multiunit firing patterns. The paper further uses this characterization to show how to rationally construct reduced operational models of neural junctions. Both uniformly proportional scaling and partial fragmentary representations are developed. The uniformly scaled models are better adapted to overall capacities and broader theoretical conceptualizations; the partial representations are better adapted to direct comparison with microelectrode experimentation. The characterization of information flow is based on coordinated multiunit patterns such as synfire chains or sequential configurations. The system can be applied to component parts of large composite networks including junctions with topographical patchiness and other irregularities. The characterization should be of use to anatomists, physiologists, modelers, and theorists. The theory predicts that the necessity for cooperative confluence of synaptic potentials in sending and receiving sequential configurations across topographically constrained projection fields requires the existence of functional ‘pattern modules’ within the topographical synaptology of the junction. Received: 13 April 1994/Accepted in revised form: 13 January 1995     

Valveless pumping in a fluid-filled closed elastic tube-system: one-dimensional theory with experimental validation

 An elastic rubber tube is connected with a stiffer rubber tube forming two halves of a torus and filled with water. Compressing one of the rubber tubes symmetrically and periodic at a point of asymmetry creates a remarkable unidirectional mean flow in the system. The size and the direction of the mean flow depend on the frequency of compression, the elasticity of the tubes, the compression ratio, and the type of compression with respect to time in a complicated manner. The system is modelled using a one-dimensional theory derived by averaging the Navier-Stokes equations ignoring higher order terms in a certain small quantity. The one-dimensional model is analysed partly analytically and partly numerically. A series of experiments on a physical realisation of the system are described. The theoretical findings and experimental results are compared; They show a remarkable agreement between the experiments and the predictions of the model. Frequencies at which the mean flow change direction are predicted numerically as well as analytically and the two results are compared. Received: 21 February 2002 / Revised version: 30 August 2002 / Published online: 17 January 2003 Key words or phrases: Flow – Elastic tubes – Valveless pumping – Navier-Stokes equations – Frequency dependent – One-dimensional model – Experimental validation     

Traveling wave solutions from microscopic to macroscopic chemotaxis models

In this paper, we study the existence and nonexistence of traveling wave solutions for the one-dimensional microscopic and macroscopic chemotaxis models. The microscopic model is based on the velocity jump process of Othmer et al. (SIAM J Appl Math 57:1044–1081, 1997). The macroscopic model, which can be shown to be the parabolic limit of the microscopic model, is the classical Keller–Segel model, (Keller and Segel in J Theor Biol 30:225–234; 377–380, 1971). In both models, the chemosensitivity function is given by the derivative of a potential function, Φ(v), which must be unbounded below at some point for the existence of traveling wave solutions. Thus, we consider two examples: F(v) = lnv{\Phi(v) = \ln v} and F(v) = ln[v/(1-v)]{\Phi(v) = \ln[v/(1-v)]}. The mathematical problem reduces to proving the existence or nonexistence of solutions to a nonlinear boundary value problem with variable coefficient on \mathbb R{\mathbb R}. The main purpose of this paper is to identify the relationships between the two models through their traveling waves, from which we can observe how information are lost, retained, or created during the transition from the microscopic model to the macroscopic model. Moreover, the underlying biological implications of our results are discussed.     

Travelling front solutions of a nonlocal Fisher equation

We consider a scalar reaction-diffusion equation containing a nonlocal term (an integral convolution in space) of which Fisher‘s equation is a particular case. We consider travelling wavefront solutions connecting the two uniform states of the equation. We show that if the nonlocality is sufficiently weak in a certain sense then such travelling fronts exist. We also construct expressions for the front and its evolution from initial data, showing that the main difference between our front and that of Fisher‘s equation is that for sufficiently strong nonlocality our front is non-monotone and has a very prominent hump. Received: 8 August 1999 / Revised: 3 March 2000 / Published online: 14 September 2000     

Relevance of nonlinear lumped-parameter models in the analysis of depth-EEG epileptic signals

In the field of epilepsy, the analysis of stereoelectroencephalographic (SEEG, intra-cerebral recording) signals with signal processing methods can help to better identify the epileptogenic zone, the area of the brain responsible for triggering seizures, and to better understand its organization. In order to evaluate these methods and to physiologically interpret the results they provide, we developed a model able to produce EEG signals from “organized” networks of neural populations. Starting from a neurophysiologically relevant model initially proposed by Lopes Da Silva et al. [Lopes da Silva FH, Hoek A, Smith H, Zetterberg LH (1974) Kybernetic 15: 27–37] and recently re-designed by Jansen et al. [Jansen BH, Zouridakis G, Brandt ME (1993) Biol Cybern 68: 275–283] the present study demonstrates that this model can be extended to generate spontaneous EEG signals from multiple coupled neural populations. Model parameters related to excitation, inhibition and coupling are then altered to produce epileptiform EEG signals. Results show that the qualitative behavior of the model is realistic; simulated signals resemble those recorded from different brain structures for both interictal and ictal activities. Possible exploitation of simulations in signal processing is illustrated through one example; statistical couplings between both simulated signals and real SEEG signals are estimated using nonlinear regression. Results are compared and show that, through the model, real SEEG signals can be interpreted with the aid of signal processing methods. Received: 3 January 2000 / Accepted: 24 March 2000     

An isomorphic mapping hypothesis of the grid representation

We introduce a grid cell microcircuit hypothesis. We propose the ‘grid in the world’ (evident in grid cell discharges) is generated by a ‘grid in the cortex’. This cortical grid is formed by patches of calbindin-positive pyramidal neurons in layer 2 of medial entorhinal cortex (MEC). Our isomorphic mapping hypothesis assumes three types of isomorphism: (i) metric correspondence of neural space (the two-dimensional cortical sheet) and the external two-dimensional space within patches; (ii) isomorphism between cellular connectivity matrix and firing field; (iii) isomorphism between single cell and population activity. Each patch is a grid cell lattice arranged in a two-dimensional map of space with a neural : external scale of approximately 1 : 2000 in the dorsal part of rat MEC. The lattice behaves like an excitable medium with neighbouring grid cells exciting each other. Spatial scale is implemented as an intrinsic scaling factor for neural propagation speed. This factor varies along the dorsoventral cortical axis. A connectivity scheme of the grid system is described. Head direction input specifies the direction of activity propagation. We extend the theory to neurons between grid patches and predict a rare discharge pattern (inverted grid cells) and the relative location and proportion of grid cells and spatial band cells.     

Convergence in almost periodic Fisher and Kolmogorov models

 We study convergence of positive solutions for almost periodic reaction diffusion equations of Fisher or Kolmogorov type. It is proved that under suitable conditions every positive solution is asymptotically almost periodic. Moreover, all positive almost periodic solutions are harmonic and uniformly stable, and if one of them is spatially homogeneous, then so are others. The existence of an almost periodic global attractor is also discussed. Received: 11 November 1996 / Revised version: 8 January 1998