Mesoscopic neurodynamics

BioSystems 64 (2002) 119–126

Mesoscopic neurodynamics

Tamas Kiss, Peter Erdi *Department of Biophysics, KFKI Research Institute for Particle and Nuclear Physics of the Hungarian Academy of Sciences,

PO Box 49, H-1525, Budapest, Hungary

Received 24 May 2001; received in revised form 3 August 2001; accepted 12 August 2001

Abstract

An overview is given on the history, theory, and applications of mesoscopic neural models. It is shown why andhow we can describe the behavior of a large population of neurons by the method of statistical neurodynamics.Normal and epileptic propagation of cortical activity have been simulated. © 2002 Elsevier Science Ireland Ltd. Allrights reserved.

Keywords: Neurodynamics; Statistical approach; Epileptic activity

www.elsevier.com/locate/biosystems

1. Introduction

The interaction between different levels of bio-logical organization was one of the main fields ofinterest of Michael Conrad:

Biological systems are treated as percolation net-works in which processes at all scales participate.Macroscopic inputs are transduced to microphys-ical e�ents through an interlea�ed hierarchy ofstructures and processes and microphysicale�ents are amplified to control macroscopicstructures and functions. Integrity and adapta-tion are achie�ed through self-consistency dy-namics operating at all le�els of organization(Conrad, 1996).

In the same spirit, in this paper we discuss thequestions of whether, why, and how it is possibleto integrate microscopic and macroscopic levels ofneural organization by applying mesoscopic-mod-eling strategy.

Structure-based bottom-up modeling has twoextreme alternatives, namely multi-compartmentalsimulations, and simulation of networks com-posed of simple elements. There is an obvioustrade-off between these two modeling strategies.The first method is appropriate to describe theelectrogenesis and spatiotemporal propagation ofthe action potential in single cells, and in smalland moderately large networks based on data ondetailed morphology and kinetics of voltage- andcalcium-dependent ion channels. The mathemati-cal framework is the celebrated Hodgkin–Huxleymodel (Hodgkin and Huxley, 1952) supplementedwith the cable theory (Rall, 1962, 1977). Neuralsimulation software such as NEURON (Hines,1984, 1993), and GENESIS (Wilson and Bower,

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T. Kiss, P. Erdi / BioSystems 64 (2002) 119–126120

1989; Bower and Beeman, 1994) have been con-structed to simplify the efficient simulation ofneurons with branching patterns. The second ap-proach offers a computationally efficient methodfor simulating a large network of neurons wherethe details of single-cell properties are neglected.There is a series of cell models with differentlevels of abstraction. While multicompartmentalmodels take into account the spatial structure of aneuron, neural network techniques are generallybased on integrate-and-fire models. The latter is aspatially homogeneous, spike-generating device.For a review of ‘spiking neurons’, see Gerstner(1999).

There is a long tradition to try to connect the‘microscopic’ single-cell behavior to the global‘macrostate’ of the nervous system analogously tothe procedures applied in statistical physics.Global brain dynamics are handled by using con-tinuous (neural field) description instead of thenetworks of discrete nerve cells. Both determinis-tic, field-theoretic (Griffith, 1963; Seelen, 1968;Wilson and Cowan, 1973; Amari, 1983), andmore statistical approaches have been developed.

In a long series of papers starting from 1974(Ventriglia, 1974, 1988, 1990, 1994), Ventrigliaconstructed a neural kinetic theory of large-scalebrain activities. Ventriglia’s kinetic theory is astatistical field theory. He assumed two types ofentities: spatially fixed neurons and spatially prop-agating spikes (‘impulses’). In our own laboratorythis model framework was adopted and modified.

In this paper, first we discuss why it is necessaryto use mesoscopic neural modeling strategy (Sec-tion 2). Then some specific model frameworks arereviewed (Section 3). Finally, several illustrativeexamples of the potential applications are given(Section 3.1).

2. Why meso-?

Mesoscopic methods bind the local (micro-scopic) and global (macroscopic) scales. The ad-jective ‘local’ means point-like here, and ‘global’may be interpreted as the whole. Obviously, localdescriptions may have very different (and relative)meanings, even if we do not go below the single-

cell level. A single neuron, which is certainly asubmicroscopic unit relative to Freeman’s micro-scopic model, may be described by a spatiallyextended (global) model by using a multicompart-mental technique. But Freeman’s model for neu-ron populations is local, since the wholepopulation is lumped into a ‘point’. In addition tospeaking about ‘local’ and ‘global’ (Freeman andKozma, 2000), we should remember the wholehierarchy of the neural organization. A step inthis direction is the statistical theory of neuralfields for describing the global brain activities interms of interacting subfields.

While the electrophysiological results obtainedby intracellular recordings have a proper corre-sponding theory based on the Hodgkin–Huxleyequations, data deriving from the brain imagingdevices lack coherent theoretical approaches.Though it became clear that the gap betweenconventional neural modeling techniques andbrain imaging data should be narrowed (Horwitzet al., 2000), there is much to be done.

Just as collective phenomena emerging in physi-cal systems made from large number of elemen-tary components (spins, molecules, etc.) aretreated by statistical mechanics, analogously havestatistical dynamical theories of neural popula-tions been established (Beurle, 1956; Griffith,1963; Seelen, 1968; Wilson and Cowan, 1973;Amari, 1974; Ventriglia, 1974, 1988, 1994; Ingber,1982; Clark et al., 1985; Peretto, 1984, 1992).Neuronal population theories established earlierhave used oversimplified single-cell models. Oneimportant example is the lack of ability to gener-ate bursts mode. Our kinetic population model(Erdi et al., 1997; Barna et al., 1988; Grobler etal., 1998) has common principles with multicom-partmental models in that both models are builtfrom anatomical and physiological data on celltypes, network connectivities, unitary synapticfunctions. There is, however, a methodologicaldifference between the two approaches. For tworeasons, our model does not contain the details ofsingle-cell events (but benefits from the results ofsingle-cell modeling!). First, even if we believedthat without very detailed biophysical descriptionof single-cell events the understanding of the func-tional organization of the whole hippocampus

T. Kiss, P. Erdi / BioSystems 64 (2002) 119–126 121

was impossible, we could neither find sufficientelementary data to build the model, nor the large-scale and long-term simulation could be done,even by supercomputers. Second, and more im-portantly, emphatically, the statistical theory weapply is based on the hypothesis that the behaviorof large neuron populations can most adequatelybe described by the methods of statisticalneurodynamics.

3. How meso-?

In the previous section we argued for the needof large-scale long-term examinations in the fieldof realistic neural networks. As we set forth be-fore the statistical formulation of certain phenom-ena carries some extra information in quantitativerespect compared with a system of many intercon-nected microscopic subsystems. In this section,some mesoscopic methods and models for de-scribing characteristic functions of different corti-cal brain areas are reviewed.

The basic notion in the followings will be thepopulation of neurons. We assume that a neuralpopulation consists of a large number of identicalneurons that have similar properties and receiveexcitatory and inhibitory synaptic input with thesame average rate. However, in some cases (e.g.Ormutag et al., 2000) the population is defined ona more general basis, the function describing thepopulation’s state is based explicitly on an ensem-ble average of replica systems. The single neuronmodel to be incorporated into the statistical de-scription may vary from the McCulloch–Pittsneurons via integrate-and-fire models to theHodgkin–Huxley model.

Based on the notion of neural populations,Mallot and Giannakopoulos (1996) presented ahighly abstract, deterministic mean-field frame-work for modeling cortical networks. In theirmodel, the state of each population is character-ized by an intracellular potential function of aneuron located in the cortical point x and time t.This potential value is transformed into a spikerate through two types of point operations: anonlinear transfer function corresponding to thesingle neuron model and spatial density function

of the cells. Network architecture is implementedby connection layers modeling dendritic and ax-onal arborization of neurons constituting the pop-ulation. A neuron in point x from a population pon the cortex can spread its activity around apoint y on the connection layer l. Activity fromdifferent populations to a certain layer is summedat all times for every layer. The dendritic arbor ofa given population then sums the activity of ap-propriate layers determining the population’s in-tracellular potential. This model framework isprimarily used for simulation of activity dynam-ics. Interesting dynamical properties, such asmultistability and limit cycle behavior of space-in-variant layered structures are considered. Layeredcortical areas are also studied and the binocularimage representation in the ocular dominancestripes of the primary visual cortex are examinedusing this model.

Nykamp and Tranchina (2000), following thework of Knight et al. (1996), presented a paperthat deals with statistical neurodynamics based onthe population density approach. As an example,they implemented a population density model ofone hypercolumn of the visual cortex and appliedit to orientation tuning. In the model, identicalintegrate-and-fire neurons are grouped to form alarge neural population. For each population,they introduced a probability density function(PDF) that represents the distribution of neuronsover all possible states. By assuming synaptictransmission with infinite velocity, and adoptingan integrate-and-fire single cell model, the inter-acting population is characterized by a singlevariable, namely with the membrane potential.

Dynamics given by the time evolution of thePDF are based on probability conservation. Prob-ability contained in a given interval (a, b) canonly change due to probability flux across theendpoints of the interval. Probability flux in themodel results from three sources: (i) the leakageflux is due to the exponential decay of the inte-grate-and-fire neurons’ voltage towards the rest-ing potential; (ii) excitation and inhibition flux areresults of conductance change due to synapticinput; (iii) probability flux across the firingthreshold potential of neurons giving the firingrate. The partial differential equations governing


the time evolution of the PDF can efficiently besolved numerically, however, the authors showthat a diffusion approximation is more suitablefor numerical analysis. The results given by thisapproximation are good if the voltage jumps dueto synaptic conductance changes are small com-pared with the voltage interval determined by theresting and threshold potentials.

In a much earlier work, also based on theprobability density function approach, Ventrigliaconstructed a neural kinetic theory of large-scalebrain activities that he presented in a series ofpapers starting from 1974 (Ventriglia, 1974, 1982,1990, 1994) His statistical theory is based on twoentities, spatially fixed neurons and spatially prop-agating impulses. Neurons might be excitatory orinhibitory and their states are characterized bytheir subthreshold membrane potential or innerexcitation, threshold level for firing, a resting levelof inactivity state, maximum hyperpolarizationlevel, absolute refractoriness period and a synap-tic delay time. Under some conditions they emitimpulses. Neurons are grouped in populations,state of the neurons in the population is describedby the population’s PDF. Impulses move freely inspace (in the numerical implementation some ruleshould be defined due to treat the effects due tospatial discretization), and might be absorbed byneurons changing their inner excitation. Impulsesare distributed in velocity-space according to thecorresponding PDF.

Grobler, Barna, and Erdi (Grobler et al., 1998;Barna et al., 1988) extended this theory by thefollowing method, they presented a diffusion the-ory in two different senses. Both the dynamicalbehavior of neurons in their state-space is and themovements of spikes in the physical space areconsidered as diffusion process. The state-space inthe model consists of the two-dimensional spacecoordinate r for both neurons and spikes, a mem-brane potential coordinate u for all types of neu-rons, and an intracellular calcium-concentrationcoordinate � for pyramidal neurons only. Bothcell types, the inhibitory and excitatory ones aredescribed by ionic conductances specific to neu-ronal type. Instead of a fixed firing threshold, asoft firing threshold is realized by voltage-depen-dent firing probability. Absorbed spikes inducetime-dependent postsynaptic conductance changein neurons, expressed by the alpha-function.

Denoting the PDF of neurons by gs(r, u, r, t)(illustration on Fig. 1, right panel) and the PDFof spikes by f s

(�)(r, t) at time t and for populations time-evolutions are formulated by balance equa-tions. For neurons the left-hand side of Eq. (1)describes the interspike dynamics containing thedrift terms and the diffusion terms for the mem-brane potential (u) and the intracellular calciumconcentration (�) variables while the right-handside contains a sink and a source term corre-sponding to neurons returning from firing andthose starting to fire, respectively.

Fig. 1. Left, an illustration for activity propegation through fibers. Direct connections between cortical areas are realized by pipes.Activity from firing cells of the source population are fed into the pipe which transmits it to cells of the target population (strengthis given in arbitrary units). Right, time evolution of populational PDF. As an illustration a one dimensional distribution of neuralstates along the membrane potential axis is plotted against time.


�gs(r, u, �, t)�t

+�

�u(�s(r, u, �, t)gs(r, u, �, t))

+�

��(�s(r, u, �, t)gs(r, u, �, t))

−Du

2�2gs(r, u, �, t)

�u2 −D�

2�2gs(r, u, �, t)

��2

=bs(r, u, �, t)−ns(r, u, �, t) (1)

Here �s and ns describes the electric and thecalcium current to the cell, respectively. Thesefunctions are determined solely by the single-cellmodel. Generally, neurons in the interspike dy-namics receiving ample excitation—which is de-termined by the voltage-dependent firingprobability, ps(u)—emit a spike and get into re-fractory period. After spending Ts amount of timein refractoriness cells return to the interspike dy-namics into a point in phase space determined bythe functions Us and Xs. The amount of neuronsgoing to and returning from refractoriness is givenby Eqs. (2) and (3), respectively.

ns(r, u, �, t)=�ps(u)�s(r, u, �, t)gs(r, u, �, t) if �s(r, u, �, t)�00 if �s(r, u, �, t)�0

(2)

bs(r, u, �, t)

=� t

−�

dt ��

−�

du �� t

−�

d� � ns(r, u �, � �, t �)

�(u−Us(u �, � �, t �))�(�−�s(u �, � �, t �))

�(t−Ts(u �, � �, t �)) (3)

Similarly, the time evolution of the PDF ofspikes is given by Eq. (4). Its left-hand side ac-counts for drift and diffusion of spikes in space,the right-hand side for spike absorption, which isproportional to the spike number, and emission,which is proportional to the number of firingcells:

�f s�(�)(r, t)

�t+ (� s�

(�)�)f s�(�)(r, t)−

Dr

2�f s�

(�)(r, t)

= −�s� f s�(�)(r, t)

+� s�(�) ��

−�

du ��

0

d� �ns�(r, u �, � �, t) (4)

where � s�(�) is the spike propagation velocity in the

direction �, �s� and � s�(�) are the absorption and

emission coefficients, respectively.Information exchange between populations s

and s � is realized by synaptic interaction when animpulse emitted by s � is absorbed by s. Postsynap-tic conductance change is proportional to theintegral of spikes absorbed (as�s) in the pastweighted by an alpha function:

�s�s=�s�s

s�s

��

0

dt �as�s(r, t− t �)t � exp�

1−t �

s�s

�(5)

where �s�s is the maximal synaptic conductanceand s�s is the time to peak. The postsynapticconductance is used to calculate the synaptic cur-rent which is an additive term in �s(r, u, �, t).

Grobler et al. (1998) performed various numeri-cal simulations to study the behavior of themodel. Results were obtained for large popula-tions of neurons, as well as for an ‘average cell’,which is a hypothetical neuron receiving the aver-age synaptic input calculated by the statisticalapproach. They showed that decreasing inhibitionin the model can result in synchronized popula-tion activity or for minimum inhibition in fullysynchronized population burst due to recurrentexcitation of pyramidal cells. Spatial activitypropagation was also found numerically by solv-ing the time evolution equations for a 0.4×1.6mm sized hippocampal CA3 slice. Without inhibi-tion propagating burst activity was observed withhighest velocity of 0.08 m/s2.

The theory proposed by Grobler et al. (1998) iscapable of capturing spatiotemporal activity evo-lution in slices of cortical areas where synapticconnections are assumed to be randomly dis-tributed and where the long-distant specific con-nections play negligible role. To study largesystem of interacting cortical areas (where theinteraction is established by long-range, specificconnections) this model has been extended (Kiss,2000). A new term to the synaptic current in �s(r,u, �, t) has been added:

�s�s=� s�sold

+� �s�s

�s�s

��

0

dt ��

(r�)

�(r, r �)as�s(r, t− td− t �)


Fig. 2. Simulations of epileptiform activity development on the cat’s cortex. Local disinhibition of posterior cingulate cortex resultedin an overall, highly synchronized activity state.

t � exp�

1−t �

�s�s

�, (6)

where the k(r, r �) function determines the sourceand target cortical area between which informationexchange occurs. Activity produced by the sourcepopulation influences the target population after td

time delay giving account of signal propagationdelay in fibers (see Fig. 1, left panel).

The diffusion theory described above follows the‘bottom-up’ concept. It first exploits knowledgeacquired from physiological single cell studies increating the primary building blocks. Second,anatomical information on connections betweenspecific brain areas is built into the model. At thesame time this diffusion theory is a constructedtheory and not a derived one, which means that thefinal form of Eq. (1) is constructed using basicprinciples of random processes and not derivedfrom the equations describing single neurons andconnections. However, efforts are done presently toderive macroscopic equations from microscopicones and relate them to the diffusion theory.

As the diffusion theory accounts for an infinitenumber of neurons its applicability was tested.Comparing both the activity– time graph and theevolution of the PDF obtained by numerical simu-lations of the diffusion theory to the averaged resultof a network with variable size of single cells, wefound that they were in good accordance even for

relatively small networks (about 1000 cells) and asthe network size increased the deviation of betweendifferent methods decreased (data not shown).

3.1. Applications

Simulations of activity propagation in the felinecerebral cortex under normal and locally disinhib-ited conditions has been performed using a modelbuilt on the last described one. Connections be-tween cortical areas were taken from the databasereported by Scannell et al. (1995). Single-cell mod-ules constituting the basic elements of neural pop-ulations are described by their membrane potential.In simulations, PDFs of neuron’s alone were takeninto account, spike propagation were modeled byusing direct connections only.

Simulations show that the probability of over-synchronization and epileptiform populationalburst activity in the disinhibited brain area isincreased as inhibition decreased. The nontrivialspread of activity from the initially bursting areathrough long-range connections is shown on Fig.2.

To have a better understanding of the mecha-nisms underlying epileptiform activity developmentwe turned our attention to simulating single corticalareas. We introduced synaptic depletion andrefilling into the synaptic conductance equations


Fig. 3. Simulation of epliepsy protocols. Electrical activity following treatment by bicucullin (left panel), and in magnesium-freesolution (right panel).

through a multiplicative resource factor which isdescribed by Eq. (7).r(t)=

��

1−exp�

−t

�−� �

t

t�=0

�(t �− tfiring) if r(t= tfiring)��

1−exp�

−t

�− �

t

t�=0

r(t �)�(t �− tfiring) if r(t= tfiting)��,

(7)

where � is the size of one quantum of transmittersubstance and the refilling time constant.

Two kinds of epilepsy protocols were examined,motivated by experiments of Doczy et al.(1999). First, we demonstrated that the treatmentby the GABAA blocker bicucullin which is re-garded as an (null) in vitro model of focalcortical epileptogenesis, evokes quasi-periodic,large amplitude population bursts with inter-burst intervals about 10�4 s (Fig. 3, leftpanel). Second, the effects of magnesium-free so-lutions were simulated by assuming enhanced re-current excitatory connections between pyramidalcells of the third layer as shown on Fig. 3, rightpanel.

We found that although both these (null)in vitro models evoked similar epileptiformelectrical activity in simulated slices characteristictimings and the fine structure of bursts were dif-ferent.

4. Conclusion

Our intention was to prove that mesoscopicneurodynamics is a necessary mathematical deviceto connect microscopic and macroscopic spatialand temporal scales. Specifically a scale-invarianttheory (and software tool) was developed, whichgives the possibility to simulate the statistical be-havior of large neural populations, and syn-chronously to monitor the behavior of an‘average’ single cell. Several illustrative applica-tions have been given.

Acknowledgements

This work was supported by the National Sci-ence Research Council (OTKA T 025500) and bythe Academic Research Grant (AKP 2001-110).

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Mesoscopic neurodynamics