Solitary Waves of Integrate and Fire Neural Fields - CiteSeerX
Solitary Waves of Integrate and Fire Neural Fields David HORN and Irit OPHER School of Physics and Astronomy Raymond and Beverly Sackler Faculty of Exact Sciences Tel Aviv University, Tel Aviv 69978, Israel [email protected]
[email protected]
February 4, 1997 Abstract Arrays of interacting identical neurons can develop coherent ring patterns, such as moving stripes that have been suggested as possible explanations of hallucinatory phenomena. Other known formations include rotating spirals and expanding concentric rings. We obtain all of them using a novel two variable description of integrate and re neurons that allows for a continuum formulation of neural elds. One of these variables distinguishes between the two di erent states of refractoriness and depolarization and acquires topological meaning when it is turned into a eld. Hence it leads to a topologic characterization of the ensuing solitary waves, or excitons. They are limited to point-like excitations on a line and linear excitations, including all the examples quoted above, on a two-dimensional surface. A moving patch of ring activity is not an allowed solitary wave on our neural surface. Only the presence of strong inhomogeneity that destroys the neural eld continuity, allows for the appearance of patchy incoherent ring patterns driven by excitatory interactions.
1 Introduction Can one construct a self consistent description of neuronal tissue on the mm scale? If so, can one use for this description the same variables that characterize a single neuron? The answers to these questions are not obvious. Assuming they are armative we are led to a theory of neural elds, describing the characteristic features of neurons at a given location and speci c time. This necessitates continuity of neural variables, which may be quite natural in view of the large overlap between neighboring neurons and the functional maps observed in di erent cortical areas showing that nearby stimuli a ect neighboring neurons. A model of neuronal tissue composed of two aggregates (excitatory and inhibitory) has been proposed long ago by Wilson and Cowan (1973). They described three different dynamical modes that correspond to di erent forms of connectivity. One of their interesting observations is the formation of pairs of unattenuated traveling waves that move in opposite directions as a result of a localized input. The propagation velocity depends on the interactions and on the amount of disinhibition in the network. Building on this approach, Amari (1977) has analyzed pattern formation in continuum distributions of neurons, or neural elds. He pointed out the possibility of local excitation solutions, and derived the conditions for their formation in one dimension. Ermentrout and Cowan (1979) have studied layers of excitatory and inhibitory neurons interacting with one another in two dimensions, and obtained the formations of moving stripes, or rolls, hexagonal lattice patterns and other forms. They pointed out that if their model is applied to V1, it can provide an explanation of drug-induced visual hallucinations (Kluver, 1967), relying on the retinocortical map interpretation 1
of V1. A simpler derivation of this result is provided by Cowan (1985), who considers a single type of neural eld on a two-dimensional manifold using a DOG (or \mexican hat") interaction with close-by strong excitation surrounded by weak inhibition. Similar questions were recently investigated by Fohlmeister et al: (1995), who have structured their neural layer according to the spike response model of Gerstner and van Hemmen (1992). Their simulations exhibit stripe formations, rotating spirals, expanding concentric rings and collective bursts. The simulations of Fohlmeister et al: (1995) are an example of how the advent of large scale computations allows one to attack elaborate neuronal models in the simulation of cortical tissues. This point was made by Hill and Villa (1994) who have studied two- dimensional arrays of 10,000 neurons and obtained ring patterns in the form of clusters or patches. Moving patches were also observed by Usher et al: (1994), who investigated a plane of integrate and re neurons with DOG interactions. This brings us to an interesting question. To what extent can one expect calculations on a grid of 100 100 neurons to re ect the behavior of a neural tissue that contains several orders of magnitude more neurons? Clearly, if the two questions raised at the beginning of this paper are answered in the armative there is a good chance to obtain meaningful results. In this paper we will characterize allowed excitation formations that obey continuity requirements, and show what happens when continuity is broken. Some of the structures found in the di erent investigations move in a fashion that conserves their general form unless they hit and merge or destroy one another. This is reminiscent of the particle property of solitons, that are known to arise in nonlinear systems (see, e.g. Newell, 1985). Yet some di erences exist. In particular, the coherent ring structures annihilate during collision rather than stay intact. Therefore, 2
an appropriate characterization is that of solitary waves (Meron, 1992). We propose using the term excitons to describe moving solitary waves in an excitable medium. Our present study is geared towards investigating these structures. We will introduce a topological description that will help us identify the type of coherent structures that are allowed under our assumptions. In particular, we will see that an object with the topological characteristics of a moving circle (patch of ring activity) is not an allowed solitary wave on a two dimensional manifold.
2 Integrate-and-Fire Neurons Integrate and re (I &F ) neurons have been chosen for simulating large systems of interacting neurons by many authors (e.g. Usher et al: , 1994). For our purpose we need a formulation of this system in which the basic variables are continuous and di erentiable. We use two such variables: v, which is a subthreshold potential, and m which distinguishes between two di erent modes in the dynamics of the single neuron, the active depolarization mode and the inactive refractory period.
v_ = ?kv + + cmv + mI
(1)
m_ = ?m + (m ? v)
(2)
(x) is the Heavyside step function. The neuron is in uenced by a constant external input I , which is absent in the absolute refractory period, when m = 0. Starting out with m = 1, the total time derivative of v is positive, and v follows the dynamics of a charging capacitor. Hence this represents the depolarization period of v. During all this time, since v < m, m stays unchanged. The dynamics change when v reaches the threshold that is arbitrarily set to 1. Then m decreases rapidly to zero, causing the time derivative of v to be negative, and v follows the dynamics of a discharging 3
v
1 0.5 0 0
50
100
150
200
250
50
100
150
200
250
50
100
150
200
250
m
1 0.5
v+8*f
0 0 4 2 0 0
Figure 1: The dynamics of the single I &F neuron. The upper frame displays v, the subthreshold membrane potential, as a function of time. The second frame shows m, the variable that distinguishes between the depolarization state, m = 1, and refractoriness, m = 0. In the third frame we plot v + 8f , where f is our spike pro le, to give a schematic presentation of the total cell potential. Parameters for this gure, as well as all 1D simulations, are: k = 0:015; = ?0:02; c = 0:0135; I = 0:05. capacitor. Parameters are chosen such that the time constants of the charging and discharging periods are di erent. To complete this description of an I &F neuron we need a quantity that represents the ring of the neuron. We introduce for this purpose
f = m(1 ? m)v
(3)
that vanishes at almost all times except when v arrives at the threshold and m changes from 1 to 0.1 This can serve therefore as a description of the action potential. An example of the dynamics of v and m is shown in Fig. 1. In a third frame we plot f gets a small contribution also when m changes from 0 to 1, but, since it is several orders of magnitude smaller, it does not have any computational consequences. An alternative choice of f dm can be proportional to ? dm dt (? dt ). 1
4
v + af , with a = 8, representing the total soma potential. The value of a is of no consequence in our work. It is used here for illustration purposes only. Our description is di erent from the FitzHugh-Nagumo model (FitzHugh, 1961), which is also a two-variable system. Their variables correspond to a two-dimensional projection of the Hodgkin-Huxley equations, and their nonlinearity is of third power rather than a step function. We believe that for the purpose of a description that leads to an intuitive insight it is important to use variables whose meaning is clear, as long as they can lead to a coarse reconstruction of the neurons' behavior.
3 I &F Neurons on a Line At this point we introduce interactions between the neurons, and see what kind of activity patterns emerge. As we are dealing with pulse coupling, an interaction is evoked by spiking of other neurons, with or without delays. Thus, Eq. 1 is replaced by: v_i = ?kvi + + cmivi + mi(I + j Wij fj ) (4) where i = 1; ; N represents the location of the neuron. Clearly the behavior of such a system of interacting neurons is strongly dependent on the property of the interaction matrix W . There are indications in V1 that excitations occur between neurons that lie close-by (Marlin, Douglas and Cynader, 1991) and inhibition is the rule further out (Worgotter, Niebur and Koch, 1991), leading to a type of centersurround pattern of interactions. To be even closer to biological reality, we should introduce di erent descriptions for excitatory (pyramidal) neurons and inhibitory interneurons. For simplicity of presentation, we consider only one set of neurons undergoing both types of interactions. We may think of the neurons as pyramidal 5
f
m
t
t
x
x
Figure 2: Creation, annihilation and motion of excitons in the x ? t plane. For xed x, they (black lines, left frame) appear on the border between m = 1 (black) and m = 0 (white) areas, shown in the right frame. cells, with inhibition being an e ective interaction. In Fig. 2 we show a spatio-temporal pattern that emerges from such a system. We assume here that the neurons are located on a line with open boundary conditions. The interaction is assumed to be simultaneous and to have a nite width on this line. In all our simulations we start with random initial conditions for vi while all mi = 1. I has the same constant value for all neurons. After some transitional period the system settles onto a spatio-temporal pattern that is a limit cycle. As the interaction strength is being increased, the behavior of vi and mi becomes a smooth function of the location i of the neuron. When this happens, we may replace vi and mi by continuous elds v(x; t) and m(x; t). Our problem may then be rede ned by a set of equations for continuous variables. For example eq. 4 becomes: dv = ?kv + + cmv + m(I + Z w(x; y)f (y)dy) (5) dt and eqs. 2 and 3 remain valid for the continuous elds. Once we are in the continuum regime, the number of neuronal units that we use in our simulations is immaterial. They represent a discretized approximation to a 6
continuum system. We have then a global description of a neural tissue in terms of two continuous variables. f (x; t) represents a volley of action potentials originated at time t by somas of neurons located at x. This is usually regarded as the interesting object to look at. As we will see, it forms one of the borderlines of regions of m in (x; t) spacetime. We propose looking at m(x; t) in order to understand the emergent behavior of this system. Looking again at Fig. 2 we see characteristic solitary wave behavior. From time to time, pairs of excitons are created. In each pair one moves to the right and the other to the left. The right moving exciton of the left pair collides with the left moving exciton of the right pair and they annihilate each other. That this has to be the case follows from the fact that ring occurs when an m = 1 neighborhood turns into an m = 0 one. In the continuum limit m(x; t) forms continuous regions with values close to either 1 or 0, thus providing us with a natural topologic argument: The border line in (x; t) can represent either a moving exciton or the creation or annihilation of a pair of excitons. Fig. 3 shows an example in which we close the line of neurons to form a ring, i.e. we use periodic, rather than open, boundary conditions. In this example we observe continuous motion of two excitons around the ring. The slope of the diagonal is determined by the propagation velocity of the exciton. As in other models of neural excitation (e.g. Wilson and Cowan, 1973), the velocity depends on the interaction matrix Wij . In our model stronger excitation leads to higher velocities (i.e. milder slopes), and the span of interactions determines the number of excitons that coexist.
7
1
−−−−>−
t
−−−−>−
0
x
x
Figure 3: Two excitons propagating along a ring. The left frame shows the ring in the x ? t plane. The right frame shows m(x) (solid line) and f (x) (dash-dotted line) at a xed time step (f was rescaled by a factor of 3 for the purpose of illustration). The arrows designate the direction of propagation. Note that for xed x ring occurs when m = 1 turns into m = 0. For xed t, the m = 0 region lies behind the moving exciton.
4 Excitons in Two Space Dimension Let us turn now to two dimensional space, and investigate a square grid of I &F neurons. We will use either DOG interactions, Wij = CE exp(?d2ij =dE ) ? CI exp(?d2ij =dI ), where dij represents the euclidean distance between two neurons, or short range excitatory connections. As will be demonstrated below, we obtain solitary waves that are lines or curves. This is to be expected since they should occur at the boundaries of m = 0 and m = 1 regions, which, in two dimensions, are one dimensional structures. We nd a variety of such ring patterns, depending on the span and strength of the interactions. Isolated arcs are a pattern typical of large range interactions with (relatively) strong inhibition. An example is shown in Fig. 4. Less inhibition leads to larger structures, as we can see in Fig. 5. 8
f
m
y
y
x
x
Figure 4: Small arcs obtained in the case of strong inhibition. CE = 0:4; CI = 0:12; dE = 5; dI = 40, restricted to a 20 20 area around each neuron on a 60 60 grid. The left frame shows the excitons in the form of small arcs that are fronts of moving m patches shown in the right frame. This gure, as well as all other 2D simulations, uses the parameters k = 0:45; c = 0:35; = ?0:09; I = 0:29:
y
x
Figure 5: A periodic pattern of activity obtained on a 90 90 spatial grid, using long range interactions: CE = 0:2; CI = 0:02; dE = 15; dI = 100. The emerging pattern is periodic in time, and consists of several propagating waves that interact. It is evident that the activity begins at the corners of the grid. This is due to the open boundary conditions. They cause the corner neurons to 9
be the least inhibited, thus rising rst and exciting their neighbors. Under similar interactions using periodic boundary conditions we get propagating parallel stripes, the cortical patterns that Ermentrout and Cowan (1979) suggested as V1 excitations corresponding to the spiral retinal forms that appear in hallucinations. Spirals and expanding rings are well known solitary wave formations (Meron, 1992) that are obtained also here. The former are displayed in Fig. 6 and the latter in Fig. 7. Both formations are often encountered in 2-d arrays of I &F neurons (e.g. Jung and Mayer-Kress, 1995, Milton, Chu and Cowan, 1993). a
b
y
y
x
x
Figure 6: Rotating spirals obtained on a 60 60 grid using DOG interactions: CE = 0:4; CI = 0:08; dE = 5; dI = 40, restricted to a 20 20 area around each neuron. The left frame (a) appears prior to the right frame (b) in the simulation. The example of expanding rings is obtained by keeping only few nearest neighbors in the interaction. The number of expanding rings is inversely related to the span of the interactions. In the example shown in Fig. 7, two spontaneous foci of excitation form expanding rings, or target formations. They are displayed in three consecutive time steps. Note that ring exists only at the boundary between m = 0 and m = 1 areas. This property is responsible for the vanishing of two ring fronts that collide, 10
because, after collision, there remains only a single m = 0 area, formed by the merger of the two former m = 0 areas. m
y
y
m
m
y
x
x
x
f
f
f
y
y
y
x
x
x
Figure 7: Expanding rings formed on a 6060 grid. The top frames show the m elds at three consecutive time steps. The bottom frames show the corresponding coherent ring patterns. In this simulation we used excitatory interactions only, coupling each neuron with its 8 neighbors with an amplitude of 0:3.
5 The Topologic Constraint The assumption that continuous neural elds can be employed leads to very strong constraints. m(~x; t) is a continuous function on the D dimensional manifold R 3 ~x. Let us denote regions in which m 12 by S1 and regions where m < 21 by S0. Clearly S0 + S1 = R. We have seen in Fig. 1 that, in practice, m is very close to 1 throughout S1, and very close to 0 throughout S0. We have discussed in this paper moving solutions of ring patterns, which we called excitons. In all these solutions 11
the regions S1 and S0 change continuously with time. Since ring can occur only when a neuron switches from a state in S1 to one in S0, excitons are restricted to domains that lie on boundaries between these two regions. Hence the dimension of has to be D ? 1. The situation is di erent for standing wave solutions. These are solutions that vary with time, sometimes in a quite complicated fashion, but do not move around in space. They are high order limit cycles. One encounters then a situation where a whole S1 region can switch into S0. This will lead to a spontaneous excitation over a region of dimension D. Starting with random initial conditions we do not generate in general such solutions. However, for regular initial conditions, e.g. a checkerboard pattern of v = 1 and 0, such solutions emerge. The typical behavior in this case shows separation of space into two parts de ned by the checkerboard pattern of initial conditions. While there is ring in one part, there is none in the other, and vice versa. The dynamics within each part can be rather complex, and does not necessarily retain all the symmetry properties that were present in the initial conditions. Since such solutions are not generated by random initial conditions we conclude that they occupy a negligible fraction of the space of dynamical attractors. The stability of a solution can be tested by seeing how it behaves in the presence of noise. Adding uctuations to I , both in space and time, we can test the stability of the di erent solutions mentioned so far. The general pattern of behavior that we nd is that, for small perturbations, excitons change to a small extent, while standing wave solutions disappear. This is to be expected in view of the fact that the standing wave solutions required especially regular initial conditions. Continuity of v and m is an outcome of the excitatory interactions of neurons with their neighborhoods. We have seen it in our simulations, for both the one and 12
two dimensional manifolds. It is a re ection of a well known property of clusters of I &F units: excitation without delays leads to coherence of ring (Mirrollo and Strogatz, 1990). We may then argue that the topologic rule will hold for all I &F systems, even ones where the eld m does not appear but v is reset to 0 after ring. The interactions lead to continuity of v. Then, even in the absence of m, there exists always a relative refractory period, when v is small, e.g. v < , in which excitations of neighboring neurons are insucient to drive v over the threshold in the next time step. In that case we can classify the manifold into S1 and S0 regions according to whether v is larger or smaller than . This leads to the same topologic consequences as the ones described above for our system.
6 Discussion Refractoriness is a basic property of neurons. We have embodied it in our formulation in an explicit fashion that turned into a topologic constraint for a theory of I &F neural elds. The important consequence is that coherent ring patterns that are obtained as solitary waves in our theory have a dimension that is smaller by one unit from that of the manifold to which they are attached. Thus we obtain point like excitons for neural elds on a line or ring, and linear ring fronts on a two dimensional surface. The system that we discussed was always under the in uence of some (usually constant) input I . Therefore it formed a coupled oscillatory system. Alternatively one may study a dissipative system of neural elds. This is an excitable medium which, in the absence of any input, will not generate excitons. However, given some initial conditions, or some local input(Chu, Milton and Cowan, 1994), it can generate target 13
fomations and spiral waves. The latter are fed by the strong interactions between the neurons. If the interactions are not strong enough, these excitons will die out. In the case of weak interactions it helps to use noisy inputs (Jung and Mayer-Kress, 1995). The latter lead to background random activity that helps to maintain the coherent formation of spirals. Sometimes there is an isotropy breaking element in the network, such as random connections or noise, that is responsible for the abundance of spiral solutions. However, spirals can be obtained also in the absence of such elements, as is the case in our work. The details of what types of dynamic attractors are dominant depend on the type of network that one studies. Nonetheless, the refractory nature of I &F neurons guarantees that the topologic rule holds for all coherent phenomena. Once one arranges identical I &F neurons on a manifold with appropriate interactions, the continuity property of neural elds follows. These continuous elds lead to coherent neural ring of the type characterized by the excitons that are described in this work. Coherence is also maintained when we add synaptic delays that are proportional to the distance between neurons. One may now ask what happens if the continuity is explicitly broken, e.g. by strong noise in the input or by randomness in the synaptic connections. What we would expect in this case is that the DOG interactions specify the resulting behavior of the system. This is, indeed, the case, as demonstrated in Fig. 8: The resulting ring behavior is quite irregular, yet it has a patchy character with a typical length scale that is of the order of the range of excitatory interactions. We believe that this is the explanation for the moving patches of activity reported in other simulations (e.g. Hill and Villa, 1994, Usher et al: 1994). They are incoherent phenomena, emerging in models with randomly distributed radial connections, as in Hill and Villa (1994) and Usher et al: (1994). A coherent moving patch of ring activity is forbidden on account of the continuity requirement
14
and refractoriness.2 a
b
y
y
x
x
Figure 8: Incoherent ring patterns for (a) high variability of synaptic connections, or (b) noisy input. In (a) we have multiplied 75 percent of all synapses by a random gaussian component (mean=1., s.d=3.) that stays constant in time. In (b) we have employed a noisy input that varies in space and time (mean=0.29, s.d.=0.25). In both frames the ring patterns are no longer excitons. We can see the formation of small clusters of ring neurons. The typical length of these patches is of the order of the span of excitatory interactions. This is a manifestation of the dominance of interactions in determining the spatial behavior in the absence of continuity that imposes the topologic constraint. We learn therefore that our model embodies two competing factors. The DOG interactions tend to produce patchy ring patterns, but the ensuing coherence of identical neurons leads to the formation of one dimensional excitons on a two dimensional manifold. If, however, strong uctuations exist, i.e. the neurons can no longer A possible exception is the case of bursting neurons. In this case sustained activity can be maintained for a short while even when the potential v reaches threshold. Hence the ring fronts can acquire some width. The e ect depends on the relation between the duration of the burst and the velocity of the exciton. 2
15
be described by homogeneous physiological and geometrical properties, the resulting patterns of ring activity will be determined by the form of the interactions. Abbott & van Vreeswijk (1993) have studied the conditions under which an I &F model with all to all coupling allows for stable asynchronous solutions. They concluded that if the interactions are instantaneous, the asynchronous state is unstable in the absence of noise. In other words, under these conditions the system is synchronous. This obviously agrees with our results. However, they found that when the synaptic interactions have various temporal structures asynchronous states can be stable. We, on the other hand, continue to obtain coherent solutions when we introduce delays in the synaptic interactions where the delays are proportional to distance. Clearly these two I &F models di er in both their geometrical and temporal structure. Our conclusions are limited to homogenous I &F structures with suitable interactions that lead to coherent behaviour. Are there situations where coherent ring activity exists in neuronal tissue? If the explanation of hallucinatory phenomena (Ermentrout and Cowan, 1979) is correct, then this is expected to be the case. It could be proved experimentally through optical imaging of V1 under appropriate pharmacological conditions. Other abnormal brain activities, such as epileptic seizures, could also fall into the category of coherent ring patterns. Does coherence occur also under normal functioning conditions? Arieli et al: (1996) have reported interesting spatiotemporal evoked activity in areas 17 and 18 in cat. Do the underlying neurons re coherently? Presumably the thalamo-cortical spindle waves that might be generated by the reticular thalamic nucleus (Golomb, Wang and Rinzel, 1994, Contreras and Steriade, 1996) can be a good example of coherent activity. Another example could be the synchronous bursts of activity that propagate as wave fronts in retinal ganglion cells of neonatal mammals (Meister et al: , 1991, Wong, 1993). It has been suggested that these waves play 16
an important role in the formation of ocular dominance layers in the LGN (Meister et al: , 1991). It would be interesting to have a systematic study of neuronal tissue on the mm scale in di erent areas and under di erent conditions, and learn if and when Nature makes use of continuous neural elds.
References Abbott, L.F. and van Vreeswijk, C. 1993. Asynchronous states in networks od pulse-coupled oscillators. Phys. Rev. E 48, 1483{1490. Amari, S. 1977. Dynamics of pattern formation in lateral- inhibition type neural elds. Biol. Cyb. 22, 77{87. Arieli, A., Sterkin A., Grinvlad A. and Aertsen A. 1996. Dynamics of ongoing activity: explanation of the large variability in evoked responses. Science 273, 1868{1871. Chu, P.S., Milton, J.G. and Cowan, J.D. 1994. Connectivity and the dynamics of integrateand- re neural networks. Int. J. of Bifurcation and Chaos 4, 237{243. Contreras, D. and Steriade, M. 1996. Spindle oscillation in cats: the role of corticothalamic feedback in a thalamically generated rhythm. J. of Physiology 490, 159{179. Cowan, J.D. 1985. What do drug induced visual hallucinations tell us about the brain? In Synaptic Modi cation, Neuron Selectivity, and Nervous System Organization, Levy WB Anderson JA and Lehmkuhle S, eds., pp.223{241. Lawrence Erlbaum, Hillsdale, NJ. Ermentrout, G.B. and Cowan J.D. 1979. A mathematical theory of visual hallucination patterns.Biol. Cyb. 34, 136{150. FitzHugh, R. 1961. Impulses and physiological states in theoretical models of nerve membrane. Biophy. J. 1, 445{466. Fohlmeister, C., Gerstner, W., Ritz, R. and van Hemmen, J. L. 1995. Spontaneous excitation in the visual cortex: stripes, spirals, rings and collective bursts. Neural Comp. 7, 905{914. Gerstner, W. and van Hemmen, J.L. 1992. Associative memory in a network of spiking
17
neurons. Network 3, 139{164. Golomb, D., Wang, X.J. and Rinzel, J. 1994. Synchronization properties of Spindle oscillations in a thalamic reticular nucleus model. J. of Neurophys. 72, 1109{1126. Hill, S.L. and Villa, A.E.P. 1994. Global spatio-temporal activity in uenced by local kinetics in a simulated \cortical" neural network. In:Supercomputing in brain research: From tomography to neural networks / workshop on supercomputing in brain research. Eds: H.J. Herrmann, D.E. Wolf and E. Poppel. World Scienti c. Jung, P. and Mayer-Kress, G. 1995. Noise controlled spiral growth in excitable media. Chaos, 5, 458{462. Kluver, H. 1967. Mescal and mechanisms of hallucinations. University of Chicago Press. Marlin, S.G., Douglas, R. M., and Cynader, M.S., 1991. Position-speci c adaptation in simple cell receptive elds of the cat striate cortex. Journal of Neurophysiology 66(5), 1769{1784. Meron, E. 1992. Pattern formation in excitable media. Physics Reports 218, 1{66. Meister M., Wong R.O.L., Denis A.B. and Shatz C.J. 1991. Synchronous bursts of action potentials in ganglion cells of the developing mammalian retina. Science 252, 939{943. Milton, J.G., Chu, P.H. and Cowan, J.D. 1993. Spiral waves in integrate-and- re neural networks. in Advances in Neural Information Processing Systems (eds. Hanson, S.J., Cowan, J.P. and Giles, C.L.) 5, 1001{1007. Mirrollo, R.E. and Strogatz, S.H. 1990. Synchronization of pulse-coupled biological oscillators, SIAM Jour. of App. Math. 50, 1645{1662. Newell, A. C. 1985. Solitons in Mathematics and Physics. Society for Industrial and Applied Mathematics. Philadelphia, PA. Usher, M., Stemmler, M., Koch, C. and Olami., Z. 1994. Network ampli cation of local
uctuations causes high spike rate variability, fractal ring patterns and oscillatory local eld potentials. Neural Comp. 6, 795{836. Wilson, H.R. and Cowan, J.D. 1973. A mathematical theory of the functional dynamics of
18
cortical and thalamic nervous tissue. Kybernetik 13, 55{80. Wong R.O.L. 1993. The role of spatio-temporal ring patterns in neuronal development of sensory systems. Curr. Op. Neurobio 3, 595{601. Worgotter, F., Niebur, E. and Koch, C. 1991. Isotropic connections generate functional asymmetrical behavior in visual cortical cells. J. Neurophysiol. 66(2), 444{459.
19
[email protected]
February 4, 1997 Abstract Arrays of interacting identical neurons can develop coherent ring patterns, such as moving stripes that have been suggested as possible explanations of hallucinatory phenomena. Other known formations include rotating spirals and expanding concentric rings. We obtain all of them using a novel two variable description of integrate and re neurons that allows for a continuum formulation of neural elds. One of these variables distinguishes between the two di erent states of refractoriness and depolarization and acquires topological meaning when it is turned into a eld. Hence it leads to a topologic characterization of the ensuing solitary waves, or excitons. They are limited to point-like excitations on a line and linear excitations, including all the examples quoted above, on a two-dimensional surface. A moving patch of ring activity is not an allowed solitary wave on our neural surface. Only the presence of strong inhomogeneity that destroys the neural eld continuity, allows for the appearance of patchy incoherent ring patterns driven by excitatory interactions.
1 Introduction Can one construct a self consistent description of neuronal tissue on the mm scale? If so, can one use for this description the same variables that characterize a single neuron? The answers to these questions are not obvious. Assuming they are armative we are led to a theory of neural elds, describing the characteristic features of neurons at a given location and speci c time. This necessitates continuity of neural variables, which may be quite natural in view of the large overlap between neighboring neurons and the functional maps observed in di erent cortical areas showing that nearby stimuli a ect neighboring neurons. A model of neuronal tissue composed of two aggregates (excitatory and inhibitory) has been proposed long ago by Wilson and Cowan (1973). They described three different dynamical modes that correspond to di erent forms of connectivity. One of their interesting observations is the formation of pairs of unattenuated traveling waves that move in opposite directions as a result of a localized input. The propagation velocity depends on the interactions and on the amount of disinhibition in the network. Building on this approach, Amari (1977) has analyzed pattern formation in continuum distributions of neurons, or neural elds. He pointed out the possibility of local excitation solutions, and derived the conditions for their formation in one dimension. Ermentrout and Cowan (1979) have studied layers of excitatory and inhibitory neurons interacting with one another in two dimensions, and obtained the formations of moving stripes, or rolls, hexagonal lattice patterns and other forms. They pointed out that if their model is applied to V1, it can provide an explanation of drug-induced visual hallucinations (Kluver, 1967), relying on the retinocortical map interpretation 1
of V1. A simpler derivation of this result is provided by Cowan (1985), who considers a single type of neural eld on a two-dimensional manifold using a DOG (or \mexican hat") interaction with close-by strong excitation surrounded by weak inhibition. Similar questions were recently investigated by Fohlmeister et al: (1995), who have structured their neural layer according to the spike response model of Gerstner and van Hemmen (1992). Their simulations exhibit stripe formations, rotating spirals, expanding concentric rings and collective bursts. The simulations of Fohlmeister et al: (1995) are an example of how the advent of large scale computations allows one to attack elaborate neuronal models in the simulation of cortical tissues. This point was made by Hill and Villa (1994) who have studied two- dimensional arrays of 10,000 neurons and obtained ring patterns in the form of clusters or patches. Moving patches were also observed by Usher et al: (1994), who investigated a plane of integrate and re neurons with DOG interactions. This brings us to an interesting question. To what extent can one expect calculations on a grid of 100 100 neurons to re ect the behavior of a neural tissue that contains several orders of magnitude more neurons? Clearly, if the two questions raised at the beginning of this paper are answered in the armative there is a good chance to obtain meaningful results. In this paper we will characterize allowed excitation formations that obey continuity requirements, and show what happens when continuity is broken. Some of the structures found in the di erent investigations move in a fashion that conserves their general form unless they hit and merge or destroy one another. This is reminiscent of the particle property of solitons, that are known to arise in nonlinear systems (see, e.g. Newell, 1985). Yet some di erences exist. In particular, the coherent ring structures annihilate during collision rather than stay intact. Therefore, 2
an appropriate characterization is that of solitary waves (Meron, 1992). We propose using the term excitons to describe moving solitary waves in an excitable medium. Our present study is geared towards investigating these structures. We will introduce a topological description that will help us identify the type of coherent structures that are allowed under our assumptions. In particular, we will see that an object with the topological characteristics of a moving circle (patch of ring activity) is not an allowed solitary wave on a two dimensional manifold.
2 Integrate-and-Fire Neurons Integrate and re (I &F ) neurons have been chosen for simulating large systems of interacting neurons by many authors (e.g. Usher et al: , 1994). For our purpose we need a formulation of this system in which the basic variables are continuous and di erentiable. We use two such variables: v, which is a subthreshold potential, and m which distinguishes between two di erent modes in the dynamics of the single neuron, the active depolarization mode and the inactive refractory period.
v_ = ?kv + + cmv + mI
(1)
m_ = ?m + (m ? v)
(2)
(x) is the Heavyside step function. The neuron is in uenced by a constant external input I , which is absent in the absolute refractory period, when m = 0. Starting out with m = 1, the total time derivative of v is positive, and v follows the dynamics of a charging capacitor. Hence this represents the depolarization period of v. During all this time, since v < m, m stays unchanged. The dynamics change when v reaches the threshold that is arbitrarily set to 1. Then m decreases rapidly to zero, causing the time derivative of v to be negative, and v follows the dynamics of a discharging 3
v
1 0.5 0 0
50
100
150
200
250
50
100
150
200
250
50
100
150
200
250
m
1 0.5
v+8*f
0 0 4 2 0 0
Figure 1: The dynamics of the single I &F neuron. The upper frame displays v, the subthreshold membrane potential, as a function of time. The second frame shows m, the variable that distinguishes between the depolarization state, m = 1, and refractoriness, m = 0. In the third frame we plot v + 8f , where f is our spike pro le, to give a schematic presentation of the total cell potential. Parameters for this gure, as well as all 1D simulations, are: k = 0:015; = ?0:02; c = 0:0135; I = 0:05. capacitor. Parameters are chosen such that the time constants of the charging and discharging periods are di erent. To complete this description of an I &F neuron we need a quantity that represents the ring of the neuron. We introduce for this purpose
f = m(1 ? m)v
(3)
that vanishes at almost all times except when v arrives at the threshold and m changes from 1 to 0.1 This can serve therefore as a description of the action potential. An example of the dynamics of v and m is shown in Fig. 1. In a third frame we plot f gets a small contribution also when m changes from 0 to 1, but, since it is several orders of magnitude smaller, it does not have any computational consequences. An alternative choice of f dm can be proportional to ? dm dt (? dt ). 1
4
v + af , with a = 8, representing the total soma potential. The value of a is of no consequence in our work. It is used here for illustration purposes only. Our description is di erent from the FitzHugh-Nagumo model (FitzHugh, 1961), which is also a two-variable system. Their variables correspond to a two-dimensional projection of the Hodgkin-Huxley equations, and their nonlinearity is of third power rather than a step function. We believe that for the purpose of a description that leads to an intuitive insight it is important to use variables whose meaning is clear, as long as they can lead to a coarse reconstruction of the neurons' behavior.
3 I &F Neurons on a Line At this point we introduce interactions between the neurons, and see what kind of activity patterns emerge. As we are dealing with pulse coupling, an interaction is evoked by spiking of other neurons, with or without delays. Thus, Eq. 1 is replaced by: v_i = ?kvi + + cmivi + mi(I + j Wij fj ) (4) where i = 1; ; N represents the location of the neuron. Clearly the behavior of such a system of interacting neurons is strongly dependent on the property of the interaction matrix W . There are indications in V1 that excitations occur between neurons that lie close-by (Marlin, Douglas and Cynader, 1991) and inhibition is the rule further out (Worgotter, Niebur and Koch, 1991), leading to a type of centersurround pattern of interactions. To be even closer to biological reality, we should introduce di erent descriptions for excitatory (pyramidal) neurons and inhibitory interneurons. For simplicity of presentation, we consider only one set of neurons undergoing both types of interactions. We may think of the neurons as pyramidal 5
f
m
t
t
x
x
Figure 2: Creation, annihilation and motion of excitons in the x ? t plane. For xed x, they (black lines, left frame) appear on the border between m = 1 (black) and m = 0 (white) areas, shown in the right frame. cells, with inhibition being an e ective interaction. In Fig. 2 we show a spatio-temporal pattern that emerges from such a system. We assume here that the neurons are located on a line with open boundary conditions. The interaction is assumed to be simultaneous and to have a nite width on this line. In all our simulations we start with random initial conditions for vi while all mi = 1. I has the same constant value for all neurons. After some transitional period the system settles onto a spatio-temporal pattern that is a limit cycle. As the interaction strength is being increased, the behavior of vi and mi becomes a smooth function of the location i of the neuron. When this happens, we may replace vi and mi by continuous elds v(x; t) and m(x; t). Our problem may then be rede ned by a set of equations for continuous variables. For example eq. 4 becomes: dv = ?kv + + cmv + m(I + Z w(x; y)f (y)dy) (5) dt and eqs. 2 and 3 remain valid for the continuous elds. Once we are in the continuum regime, the number of neuronal units that we use in our simulations is immaterial. They represent a discretized approximation to a 6
continuum system. We have then a global description of a neural tissue in terms of two continuous variables. f (x; t) represents a volley of action potentials originated at time t by somas of neurons located at x. This is usually regarded as the interesting object to look at. As we will see, it forms one of the borderlines of regions of m in (x; t) spacetime. We propose looking at m(x; t) in order to understand the emergent behavior of this system. Looking again at Fig. 2 we see characteristic solitary wave behavior. From time to time, pairs of excitons are created. In each pair one moves to the right and the other to the left. The right moving exciton of the left pair collides with the left moving exciton of the right pair and they annihilate each other. That this has to be the case follows from the fact that ring occurs when an m = 1 neighborhood turns into an m = 0 one. In the continuum limit m(x; t) forms continuous regions with values close to either 1 or 0, thus providing us with a natural topologic argument: The border line in (x; t) can represent either a moving exciton or the creation or annihilation of a pair of excitons. Fig. 3 shows an example in which we close the line of neurons to form a ring, i.e. we use periodic, rather than open, boundary conditions. In this example we observe continuous motion of two excitons around the ring. The slope of the diagonal is determined by the propagation velocity of the exciton. As in other models of neural excitation (e.g. Wilson and Cowan, 1973), the velocity depends on the interaction matrix Wij . In our model stronger excitation leads to higher velocities (i.e. milder slopes), and the span of interactions determines the number of excitons that coexist.
7
1
−−−−>−
t
−−−−>−
0
x
x
Figure 3: Two excitons propagating along a ring. The left frame shows the ring in the x ? t plane. The right frame shows m(x) (solid line) and f (x) (dash-dotted line) at a xed time step (f was rescaled by a factor of 3 for the purpose of illustration). The arrows designate the direction of propagation. Note that for xed x ring occurs when m = 1 turns into m = 0. For xed t, the m = 0 region lies behind the moving exciton.
4 Excitons in Two Space Dimension Let us turn now to two dimensional space, and investigate a square grid of I &F neurons. We will use either DOG interactions, Wij = CE exp(?d2ij =dE ) ? CI exp(?d2ij =dI ), where dij represents the euclidean distance between two neurons, or short range excitatory connections. As will be demonstrated below, we obtain solitary waves that are lines or curves. This is to be expected since they should occur at the boundaries of m = 0 and m = 1 regions, which, in two dimensions, are one dimensional structures. We nd a variety of such ring patterns, depending on the span and strength of the interactions. Isolated arcs are a pattern typical of large range interactions with (relatively) strong inhibition. An example is shown in Fig. 4. Less inhibition leads to larger structures, as we can see in Fig. 5. 8
f
m
y
y
x
x
Figure 4: Small arcs obtained in the case of strong inhibition. CE = 0:4; CI = 0:12; dE = 5; dI = 40, restricted to a 20 20 area around each neuron on a 60 60 grid. The left frame shows the excitons in the form of small arcs that are fronts of moving m patches shown in the right frame. This gure, as well as all other 2D simulations, uses the parameters k = 0:45; c = 0:35; = ?0:09; I = 0:29:
y
x
Figure 5: A periodic pattern of activity obtained on a 90 90 spatial grid, using long range interactions: CE = 0:2; CI = 0:02; dE = 15; dI = 100. The emerging pattern is periodic in time, and consists of several propagating waves that interact. It is evident that the activity begins at the corners of the grid. This is due to the open boundary conditions. They cause the corner neurons to 9
be the least inhibited, thus rising rst and exciting their neighbors. Under similar interactions using periodic boundary conditions we get propagating parallel stripes, the cortical patterns that Ermentrout and Cowan (1979) suggested as V1 excitations corresponding to the spiral retinal forms that appear in hallucinations. Spirals and expanding rings are well known solitary wave formations (Meron, 1992) that are obtained also here. The former are displayed in Fig. 6 and the latter in Fig. 7. Both formations are often encountered in 2-d arrays of I &F neurons (e.g. Jung and Mayer-Kress, 1995, Milton, Chu and Cowan, 1993). a
b
y
y
x
x
Figure 6: Rotating spirals obtained on a 60 60 grid using DOG interactions: CE = 0:4; CI = 0:08; dE = 5; dI = 40, restricted to a 20 20 area around each neuron. The left frame (a) appears prior to the right frame (b) in the simulation. The example of expanding rings is obtained by keeping only few nearest neighbors in the interaction. The number of expanding rings is inversely related to the span of the interactions. In the example shown in Fig. 7, two spontaneous foci of excitation form expanding rings, or target formations. They are displayed in three consecutive time steps. Note that ring exists only at the boundary between m = 0 and m = 1 areas. This property is responsible for the vanishing of two ring fronts that collide, 10
because, after collision, there remains only a single m = 0 area, formed by the merger of the two former m = 0 areas. m
y
y
m
m
y
x
x
x
f
f
f
y
y
y
x
x
x
Figure 7: Expanding rings formed on a 6060 grid. The top frames show the m elds at three consecutive time steps. The bottom frames show the corresponding coherent ring patterns. In this simulation we used excitatory interactions only, coupling each neuron with its 8 neighbors with an amplitude of 0:3.
5 The Topologic Constraint The assumption that continuous neural elds can be employed leads to very strong constraints. m(~x; t) is a continuous function on the D dimensional manifold R 3 ~x. Let us denote regions in which m 12 by S1 and regions where m < 21 by S0. Clearly S0 + S1 = R. We have seen in Fig. 1 that, in practice, m is very close to 1 throughout S1, and very close to 0 throughout S0. We have discussed in this paper moving solutions of ring patterns, which we called excitons. In all these solutions 11
the regions S1 and S0 change continuously with time. Since ring can occur only when a neuron switches from a state in S1 to one in S0, excitons are restricted to domains that lie on boundaries between these two regions. Hence the dimension of has to be D ? 1. The situation is di erent for standing wave solutions. These are solutions that vary with time, sometimes in a quite complicated fashion, but do not move around in space. They are high order limit cycles. One encounters then a situation where a whole S1 region can switch into S0. This will lead to a spontaneous excitation over a region of dimension D. Starting with random initial conditions we do not generate in general such solutions. However, for regular initial conditions, e.g. a checkerboard pattern of v = 1 and 0, such solutions emerge. The typical behavior in this case shows separation of space into two parts de ned by the checkerboard pattern of initial conditions. While there is ring in one part, there is none in the other, and vice versa. The dynamics within each part can be rather complex, and does not necessarily retain all the symmetry properties that were present in the initial conditions. Since such solutions are not generated by random initial conditions we conclude that they occupy a negligible fraction of the space of dynamical attractors. The stability of a solution can be tested by seeing how it behaves in the presence of noise. Adding uctuations to I , both in space and time, we can test the stability of the di erent solutions mentioned so far. The general pattern of behavior that we nd is that, for small perturbations, excitons change to a small extent, while standing wave solutions disappear. This is to be expected in view of the fact that the standing wave solutions required especially regular initial conditions. Continuity of v and m is an outcome of the excitatory interactions of neurons with their neighborhoods. We have seen it in our simulations, for both the one and 12
two dimensional manifolds. It is a re ection of a well known property of clusters of I &F units: excitation without delays leads to coherence of ring (Mirrollo and Strogatz, 1990). We may then argue that the topologic rule will hold for all I &F systems, even ones where the eld m does not appear but v is reset to 0 after ring. The interactions lead to continuity of v. Then, even in the absence of m, there exists always a relative refractory period, when v is small, e.g. v < , in which excitations of neighboring neurons are insucient to drive v over the threshold in the next time step. In that case we can classify the manifold into S1 and S0 regions according to whether v is larger or smaller than . This leads to the same topologic consequences as the ones described above for our system.
6 Discussion Refractoriness is a basic property of neurons. We have embodied it in our formulation in an explicit fashion that turned into a topologic constraint for a theory of I &F neural elds. The important consequence is that coherent ring patterns that are obtained as solitary waves in our theory have a dimension that is smaller by one unit from that of the manifold to which they are attached. Thus we obtain point like excitons for neural elds on a line or ring, and linear ring fronts on a two dimensional surface. The system that we discussed was always under the in uence of some (usually constant) input I . Therefore it formed a coupled oscillatory system. Alternatively one may study a dissipative system of neural elds. This is an excitable medium which, in the absence of any input, will not generate excitons. However, given some initial conditions, or some local input(Chu, Milton and Cowan, 1994), it can generate target 13
fomations and spiral waves. The latter are fed by the strong interactions between the neurons. If the interactions are not strong enough, these excitons will die out. In the case of weak interactions it helps to use noisy inputs (Jung and Mayer-Kress, 1995). The latter lead to background random activity that helps to maintain the coherent formation of spirals. Sometimes there is an isotropy breaking element in the network, such as random connections or noise, that is responsible for the abundance of spiral solutions. However, spirals can be obtained also in the absence of such elements, as is the case in our work. The details of what types of dynamic attractors are dominant depend on the type of network that one studies. Nonetheless, the refractory nature of I &F neurons guarantees that the topologic rule holds for all coherent phenomena. Once one arranges identical I &F neurons on a manifold with appropriate interactions, the continuity property of neural elds follows. These continuous elds lead to coherent neural ring of the type characterized by the excitons that are described in this work. Coherence is also maintained when we add synaptic delays that are proportional to the distance between neurons. One may now ask what happens if the continuity is explicitly broken, e.g. by strong noise in the input or by randomness in the synaptic connections. What we would expect in this case is that the DOG interactions specify the resulting behavior of the system. This is, indeed, the case, as demonstrated in Fig. 8: The resulting ring behavior is quite irregular, yet it has a patchy character with a typical length scale that is of the order of the range of excitatory interactions. We believe that this is the explanation for the moving patches of activity reported in other simulations (e.g. Hill and Villa, 1994, Usher et al: 1994). They are incoherent phenomena, emerging in models with randomly distributed radial connections, as in Hill and Villa (1994) and Usher et al: (1994). A coherent moving patch of ring activity is forbidden on account of the continuity requirement
14
and refractoriness.2 a
b
y
y
x
x
Figure 8: Incoherent ring patterns for (a) high variability of synaptic connections, or (b) noisy input. In (a) we have multiplied 75 percent of all synapses by a random gaussian component (mean=1., s.d=3.) that stays constant in time. In (b) we have employed a noisy input that varies in space and time (mean=0.29, s.d.=0.25). In both frames the ring patterns are no longer excitons. We can see the formation of small clusters of ring neurons. The typical length of these patches is of the order of the span of excitatory interactions. This is a manifestation of the dominance of interactions in determining the spatial behavior in the absence of continuity that imposes the topologic constraint. We learn therefore that our model embodies two competing factors. The DOG interactions tend to produce patchy ring patterns, but the ensuing coherence of identical neurons leads to the formation of one dimensional excitons on a two dimensional manifold. If, however, strong uctuations exist, i.e. the neurons can no longer A possible exception is the case of bursting neurons. In this case sustained activity can be maintained for a short while even when the potential v reaches threshold. Hence the ring fronts can acquire some width. The e ect depends on the relation between the duration of the burst and the velocity of the exciton. 2
15
be described by homogeneous physiological and geometrical properties, the resulting patterns of ring activity will be determined by the form of the interactions. Abbott & van Vreeswijk (1993) have studied the conditions under which an I &F model with all to all coupling allows for stable asynchronous solutions. They concluded that if the interactions are instantaneous, the asynchronous state is unstable in the absence of noise. In other words, under these conditions the system is synchronous. This obviously agrees with our results. However, they found that when the synaptic interactions have various temporal structures asynchronous states can be stable. We, on the other hand, continue to obtain coherent solutions when we introduce delays in the synaptic interactions where the delays are proportional to distance. Clearly these two I &F models di er in both their geometrical and temporal structure. Our conclusions are limited to homogenous I &F structures with suitable interactions that lead to coherent behaviour. Are there situations where coherent ring activity exists in neuronal tissue? If the explanation of hallucinatory phenomena (Ermentrout and Cowan, 1979) is correct, then this is expected to be the case. It could be proved experimentally through optical imaging of V1 under appropriate pharmacological conditions. Other abnormal brain activities, such as epileptic seizures, could also fall into the category of coherent ring patterns. Does coherence occur also under normal functioning conditions? Arieli et al: (1996) have reported interesting spatiotemporal evoked activity in areas 17 and 18 in cat. Do the underlying neurons re coherently? Presumably the thalamo-cortical spindle waves that might be generated by the reticular thalamic nucleus (Golomb, Wang and Rinzel, 1994, Contreras and Steriade, 1996) can be a good example of coherent activity. Another example could be the synchronous bursts of activity that propagate as wave fronts in retinal ganglion cells of neonatal mammals (Meister et al: , 1991, Wong, 1993). It has been suggested that these waves play 16
an important role in the formation of ocular dominance layers in the LGN (Meister et al: , 1991). It would be interesting to have a systematic study of neuronal tissue on the mm scale in di erent areas and under di erent conditions, and learn if and when Nature makes use of continuous neural elds.
References Abbott, L.F. and van Vreeswijk, C. 1993. Asynchronous states in networks od pulse-coupled oscillators. Phys. Rev. E 48, 1483{1490. Amari, S. 1977. Dynamics of pattern formation in lateral- inhibition type neural elds. Biol. Cyb. 22, 77{87. Arieli, A., Sterkin A., Grinvlad A. and Aertsen A. 1996. Dynamics of ongoing activity: explanation of the large variability in evoked responses. Science 273, 1868{1871. Chu, P.S., Milton, J.G. and Cowan, J.D. 1994. Connectivity and the dynamics of integrateand- re neural networks. Int. J. of Bifurcation and Chaos 4, 237{243. Contreras, D. and Steriade, M. 1996. Spindle oscillation in cats: the role of corticothalamic feedback in a thalamically generated rhythm. J. of Physiology 490, 159{179. Cowan, J.D. 1985. What do drug induced visual hallucinations tell us about the brain? In Synaptic Modi cation, Neuron Selectivity, and Nervous System Organization, Levy WB Anderson JA and Lehmkuhle S, eds., pp.223{241. Lawrence Erlbaum, Hillsdale, NJ. Ermentrout, G.B. and Cowan J.D. 1979. A mathematical theory of visual hallucination patterns.Biol. Cyb. 34, 136{150. FitzHugh, R. 1961. Impulses and physiological states in theoretical models of nerve membrane. Biophy. J. 1, 445{466. Fohlmeister, C., Gerstner, W., Ritz, R. and van Hemmen, J. L. 1995. Spontaneous excitation in the visual cortex: stripes, spirals, rings and collective bursts. Neural Comp. 7, 905{914. Gerstner, W. and van Hemmen, J.L. 1992. Associative memory in a network of spiking
17
neurons. Network 3, 139{164. Golomb, D., Wang, X.J. and Rinzel, J. 1994. Synchronization properties of Spindle oscillations in a thalamic reticular nucleus model. J. of Neurophys. 72, 1109{1126. Hill, S.L. and Villa, A.E.P. 1994. Global spatio-temporal activity in uenced by local kinetics in a simulated \cortical" neural network. In:Supercomputing in brain research: From tomography to neural networks / workshop on supercomputing in brain research. Eds: H.J. Herrmann, D.E. Wolf and E. Poppel. World Scienti c. Jung, P. and Mayer-Kress, G. 1995. Noise controlled spiral growth in excitable media. Chaos, 5, 458{462. Kluver, H. 1967. Mescal and mechanisms of hallucinations. University of Chicago Press. Marlin, S.G., Douglas, R. M., and Cynader, M.S., 1991. Position-speci c adaptation in simple cell receptive elds of the cat striate cortex. Journal of Neurophysiology 66(5), 1769{1784. Meron, E. 1992. Pattern formation in excitable media. Physics Reports 218, 1{66. Meister M., Wong R.O.L., Denis A.B. and Shatz C.J. 1991. Synchronous bursts of action potentials in ganglion cells of the developing mammalian retina. Science 252, 939{943. Milton, J.G., Chu, P.H. and Cowan, J.D. 1993. Spiral waves in integrate-and- re neural networks. in Advances in Neural Information Processing Systems (eds. Hanson, S.J., Cowan, J.P. and Giles, C.L.) 5, 1001{1007. Mirrollo, R.E. and Strogatz, S.H. 1990. Synchronization of pulse-coupled biological oscillators, SIAM Jour. of App. Math. 50, 1645{1662. Newell, A. C. 1985. Solitons in Mathematics and Physics. Society for Industrial and Applied Mathematics. Philadelphia, PA. Usher, M., Stemmler, M., Koch, C. and Olami., Z. 1994. Network ampli cation of local
uctuations causes high spike rate variability, fractal ring patterns and oscillatory local eld potentials. Neural Comp. 6, 795{836. Wilson, H.R. and Cowan, J.D. 1973. A mathematical theory of the functional dynamics of
18
cortical and thalamic nervous tissue. Kybernetik 13, 55{80. Wong R.O.L. 1993. The role of spatio-temporal ring patterns in neuronal development of sensory systems. Curr. Op. Neurobio 3, 595{601. Worgotter, F., Niebur, E. and Koch, C. 1991. Isotropic connections generate functional asymmetrical behavior in visual cortical cells. J. Neurophysiol. 66(2), 444{459.
19
