neural systems in which the object of study is a region containing many cells. In the model pioneered by Wilson and. Cowan[l] the details of synaptic connections ...
A CONTINUUM MODEL OF ACTIVITY WAVES IN LAYERED NEURONAL NETWORKS : COMPUTER MODELS OF BRAIN-STEM SEIZURES PAUL KOCH Dept. of Computer Science, New York Institute of Technology Central Islip, New York 11722 GERALD LEISMAN Division of Research, New York Chiropractic College Glen Head, New York 11545 ABSTRACT We model dysfunctional brain-stem activity as nonlinear homogeneous disturbances in a structure consisting of excitatory and inhibitory layers. We identify cortical EEG traces with the time derivative of the active fraction of excitatory cells. Increase in the time delay between the layers causes onset of periodic nonlinear oscillations. Change in the biochemical state can cause rapid oscillations similar to seizures. INTRODUCTION Continuum neural dynamics[1,2,3] is an approach to large neural systems in which the object of study is a region containing many cells. In the model pioneered by Wilson and Cowan[l] the details of synaptic connections within such differential regions are invisible and connections among regions are probabilistic. Neural tissue is treated as an active medium, consisting of purely excitatory and inhibitory cells, whose material properties are to be determined by standard methods of mathematical physics. In general, active media (materials with a source of free energy) amplify small signals, and linear analysis is appropriate to describe this process only as long as the amplified disturbance (measured here as a fraction of active cells[l]) remains sufficiently small. We have previously[4] applied such a linear analysis to the lower brain, referred to f o r convenience as the "brain stem" and/or "reticular formation", which anatomically is largely separated into excitatory and inhibitory areas. This region is at once the center of conscious attention[S] and the site of origin of epileptic seizurest61. For analytical tractability the structure was simplified into two layers each consisting of cells of a single type, with a time delay in transmission of signals between the layers. It is believed[5,7] that the complex reticular circuits function
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larse part to supply s ~ c ha delay.
When the Wilson-Cowan equations are linearized and Fourier analysis is applied, it is found[3,4] that the delay is crucial in determining the linear response of the model to small input signals. Generally speaking, an increase in delay causes increased signal amplification, but the growth exhibits a discontinuity at a critical delay value T,. We have conjectured[4] that delays below Tc control the normal competition for attention among reticular areas[5], and the larger delays, which result in greatly enhanced amplification, are dysfunctional. However, it is exactly in this latter case that the linear assumption breaks down, and non-linear analysis becomes necessary. An important characteristic of the system response in this regime is its relative homogeneity. A point signal gives rise to massive activity which diffuses into adjacent regions, with almost no spatial structure in areas into which it has penetrated[4]. This feature simplifies nonlinear analysisI31, but the situation is complicated by the introduction of several new parameters that characterize the biochemical state. An exhaustive search of the resulting multidimensional parameter space is in order; we present here some preliminary results, which exhibit many features of EEG diagnostics of seizure victims. These results indicate how changes in delay time and chemical state may lead to the onset of brain-stem seizures. MATHEMATICAL FORMULATION The assumptions and derivation leading to the Wilson-Cowan equations are presented elsewhere[l,31. For present purposes, we take the dependent variables to be E(x,t) and I(x,t), respectively the fractions of cells in the excitatory and inhibitory layers at position x active at time t. When E and I are independent of space, the relevant equations have the form 133: dE/dt dI/dt
+ +
E = Se(Kee E(t) - Kie I(t)) I = Si(Kei E(t-T)
-
Kii I(t)).
Here Krs (r,s = e,i) is the connection-strength tensor, which can be assumed constant[3]. The total inhibitory delay is T, the sum of Te and Ti, respectively the delay in signal transmission from the excitatory to the inhibitory region and the delay in the return of the resulting inhibitory signal to the stimulated excitatory cells[3]. For convenience, I as defined in Equation ( 1 ) is I(t - Te). For species s ( s = e or i) Ss is the sigmoid input-output characteristic function which is taken to have the formtll:
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In Equation (2) the species parameters n and H determine respectively the size and location of the maximum slope of S. The excited fractions E and I in Equation (1) are measured with respect to species equilibrium values Q [1,3]. Examination of the equilibrium points of Equations ( 1 7 (i.e. points of zero time derivative[3]) shows that the second term in Equation ( 2 ) is in fact the equilibrium excitation level. The three parameters, which we assume to be determined by the biochemical state, are thus related by: ns Hs = h[(l - Qs)/Qsl.
(3)
Note that the species refractive recovery times111 have been omitted from Equations (1). These equations are valid only in the limit of small refractory time and its inclusion is a complication that makes no qualitative contribution to the results. In order to relate the present results to the function of the normal brain stem[4] we note that the linear form of Equations (1) is: dE/dt + E = Be E(t) - Ce I(t) (4)
dI/dt + I + Ci E(t-T) - Bi I(t), where Be = seKee, Ce = seKie Ci = SiK,i,
Bi = SiKii, ( 5 )
and s s Is= e?i\ is the slope of the species sigmoid function at zero net disturbance (i.e. equilibrium excitation): ss =
ns Qs 1 1 -
QS\,
(6)
a function of the chemical state. NUMERICAL RESULTS Equations (4) are actually the homogenous limit (wave number k approaching zero) of the more general Fourier-analyzed linear Wilson-Cowan Equations141. We introduce a "reference state" with the following parameter values: Be = 2.8, Ce Qe
=
0.08
=
6.09, Ci = 1.48, Bi = 2.7
Qi = 0.5.
(7)
For this state, we show in Figure (1) the linear growth rate from Fourier-Laplace analysis of Equation (4), as a function
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of delay time T and wave number k. The linear parameters in Equation ( 7 ) , and the ranges of the connection types have been chosen[4] so that the model can function as an attentional mechanism in the normal range of delay (here T < 0.8 decay periods approximately). Wave number is normalized in terms of the interlayer connection rangel41. The parameter regime of interest here is at large T, where the growth rate undergoes a discontinuous increase and is greatest at k = 0 (homogeneous amplification). Linear analysis also indicates that the temporal frequency of this preferentially amplified disturbance is zerot41, SO that altogether linear theory predicts growth without oscillation. The periodic behavior that we observe in many of the limits of Equation ( 1) thus comes from a nonlinear effect, viz. saturation of the sigmoid function. When the species equilibria are unbalanced, as in Equation ( 7 1 , periodicity sets in through a bifurcation that occurs with increasing T, as illustrated in Figure (2). For the given parameters, and an initial disturbance E t=o 0.0008, the bifurcation delay T is approximately .399)48, At delays below Tb, the excite2 fraction E of excitatory cells approaches a value near its upper equilibrium point E = I - Q,, and its derivative E' exhibits decreasing oscillations. However above the bifurcation delay, periodic spikes occur in E' as E oscillates between its upper and lower (E = - Qe) equilibrium points.
f
The similarity between the spikes in E' and "interictal spikes"[ 8 1 observed between epileptic seizures leads us to believe[31 that cortical EEG signals are proportional to the time derivative of the brain-stem excited fraction E. Using this assumption we are interested in determining the circumstances under which the faster oscillations that characterize seizures[8] could be observed. We note from Equations (3) - (6) that changes in the biochemical state can affect both the linear and nonlinear responses of the system through two independent parameters, conveniently ns and Q, for species s. Using the reference state we define Fns as the relative sensitivity n/nref for species s. The value 0.08 chosen in Equation (7) as the datum for the excitatory fraction E reflects a large positive species threshold (Equation ( 3 ) ) , as would be appropriate for the normal reticular competition for attentiont4,51. Increase of the equilibrium level (caused by decrease of the species threshold) increases Be (Equations (5) and (6)), a linear coefficient that tends to increase growth (Equation (4)). Although Ce also increases, the net effect is to decrease the critical delay Tc at which non-linear homogeneous growth sets in. Thus when Qe = 0.5 and Ke and Kie are kept constant (effect is to multiply Be an2 ce by 0 . 2 5 / 0 . 0 9 with
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respect to the reference state), we find that Tc < 0.24. As shown in Figure (3) the resulting E'(and EEG trace by assumption) is symmetric in the up-down direction. The frequency, however, is still rather slow. In Figure (4) we show that increasing the "sensitivity" n of the inhibitory region (Fni = 15) leads to oscillations about ten times the spike frequency, which is near the correct order of magnitude for seizures[8]. By itself, increased Fni causes both the linear and nonlinear temporal frequencies to increase, but when Q is simultaneously increased the linear frequency goes rapidfy to zero at constant delay T. The oscillations in Figure (4), at Qe = 0.5, Fni = 15 and T = 0.24 are both rapid and nonlinear. REFERENCES
I11
H.R. Wilson and J.D. Cowan, "A mathematical theory of the functional dynamics of cortical and thalamic nervous tissue:, Kybernetic, vol. 13., pp 55-80, 1973.
t21
G. Krone, H. Mallot, G. Palm and A. Schuz, "Spatiotemporal receptive fields: a dynamical model deiived f;om cortical architectonics", -Proc. Soc. Lond., vol. B226, pp 421-444, 1986.
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P. Koch and G. Leisman, "The layered neural continuum", submitted to Journal of Enq. Medicine Biol., IEEE, August 1989.
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P. Koch and G. Leisman, "Model of attentional brain function", in International Joint Conference on Neural Networks, San Diego: IEEE TAB Neural Network Committee, 1989, vol. 1, pp 775-760.
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M.E. Scheibel and A.B. Scheibel, "Anatomical basis of, attention mechanisms in vertebrate brains" , in G.C. Quarton, T. Melechuk and F.O. Schmitt (eds) The Neurosciences. A Study Program., New York: The Rockefeller Univergity Press, 1967, pp 577-602.
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B. Ralston and C. Ajmone-Marsan, "Thalamic control of certain normal and abnormal cortical rhythms", Electroenceph. and Clin. Neurophys., vol. 8, pp 559567, 1956.
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J.E. Desmond and J.W. Moore, "Adaptive timing in neural networks : the conditioned response" , Biol. Cyber , vol 58, pp 405-4i5, 1988.
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R.C. Collins and T.V. Caston, "Functional anatomy of occipital lobe seizures: an experimental study in rats", Neurology, vol. 29, pp 705-716, 1979.
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