{"title": "Spiral Waves in Integrate-and-Fire Neural Networks", "book": "Advances in Neural Information Processing Systems", "page_first": 1001, "page_last": 1006, "abstract": null, "full_text": "Spiral Waves in Integrate-and-Fire \n\nNeural Networks \n\nJohn G. Milton \n\nDepartment of Neurology \nThe University of Chicago \n\nChicago, IL 60637 \n\nPo Hsiang Chu \n\nDepartment of Computer Science \n\nDePaul University \nChicago, IL 60614 \n\nJack D. Cowan \n\nDepartment of Mathematics \nThe University of Chicago \n\nChicago, IL 60637 \n\nAbstract \n\nThe formation of propagating spiral waves is studied in a randomly \nconnected neural network composed of integrate-and-fire neurons \nwith recovery period and excitatory connections using computer \nsimulations. Network activity is initiated by periodic stimulation \nat a single point. The results suggest that spiral waves can arise in \nsuch a network via a sub-critical Hopf bifurcation. \n\n1 \n\nIntroduction \n\nIn neural networks activity propagates through populations, or layers, of neurons. \nThis propagation can be monitored as an evolution of spatial patterns of activity. \nThirty years ago, computer simulations on the spread of activity through 2-D ran(cid:173)\ndomly connected networks demonstrated that a variety of complex spatio-temporal \npatterns can be generated including target waves and spirals (Beurle, 1956, 1962; \nFarley and Clark, 1961; Farley, 1965). The networks studied by these investigators \ncorrespond to inhomogeneous excitable media in which the probability of interneu(cid:173)\nronal connectivity decreases exponentially with distance. Although travelling spiral \nwaves can readily be formed in excitable media by the introduction of non-uniform \n\n1001 \n\n\f1002 \n\nMilton, Chu, and Cowan \n\ninitial conditions (e.g. Winfree, 1987), this approach is not suitable for the study \nand classification of the dynamics associated with the onset of spiral wave forma(cid:173)\ntion. Here we show that spiral waves can \"spontaneously\" arise from target waves \nin a neural network in which activity is initiated by periodic stimulation at a sin(cid:173)\ngle point. In particular, the onset of spiral wave formation appears to occur via a \nsub-critical Hopf bifurcation. \n\n2 Methods \n\nComputer simulations were used to simulate the propagation of activity from a \ncentrally placed source in a neural network containing 100 x 100 neurons arranged \non a square lattice with excitatory interactions. At t = 0 all neurons were at rest \nexcept the source. There were free boundary conditions and all simulations were \nperformed on a SUN SPARC 1+ computer. \n\nThe network was constructed by assuming that the probability, A, of interneuronal \nconnectivity was an exponential decreasing function of distance, i.e. \n\nA = (3 exp( -air!) \n\nwhere a = 0.6, {3 = 1.5 are constants and Ir I is the euclidean interneuronal distance \n(on average each neuron makes 24 connections and '\" 1.3 connections per neuron, \ni.e. multiple connections occur). Once the connectivity was determined it remained \nfixed throughout the simulation. \n\nThe dynamics of each neuron were represented by an integrate-and-fire model pos(cid:173)\nsessing a \"leaky\" membrane potential and an absolute (1 time step) and relative \nrefractory or recovery period as described previously (Beurle, 1962; Farley, 1965; \nFarley and Clark, 1961): the membrane and threshold decay constants were, respec(cid:173)\ntively, k m = 0.3 msec- 1 , ko = 0.03 msec- 1\u2022 The time step of the network was taken \nas 1 msec and it was assumed that during this time a neuron transmits excitation \nto all other neurons connected to it. \n\n3 Results \n\nWe illustrate the dynamics of a particular network as a function of the magnitude \nof the excitatory interneuronal excitation, E, when all other parameters are fixed. \nWhen E < 0.2 no activity propagates from the central source. For 0.2 < E < 0.58 \ntarget waves regularly emanate from the centrally placed source (Figure 1a). For \nE ~ 0.58 the activity patterns, once established, persisted even when the source \nwas turned off. Complex spiral waves occurred when 0.58 < E < 0.63 (Figures \n1b-ld). In these cases spiral meandering, spiral tip break-up and the formation of \nnew spirals (some with multiple arms) occur continuously. Eventually the spirals \ntend to migrate out of the network. For E ~ 0.63 only disorganized spatial patterns \noccurred without clearly distinguishable wave fronts, except initially (Figures ie-f). \n\n\fSpiral Waves in Integrate-and-Fire Neural Networks \n\n1003 \n\nFigure 1: Representative examples of the spatial pattern of neural activity as a \nfunction of E:(a) E = 0.45, (b - e) E = 0.58 and (f) E = 0.72. Color code: gray = \nquiescent, white = activated, black = relatively refractory. See text for details. \n\n,~~~~~~~WL \n\nb \n\nc \n\n.1 \n\nD5 \n\n.07 \n\nc.:I \n0 \nZ \n~ .14 \ni:L \nCIl z \n0 \n~ :::> \nU.I z \nz \n0 \n~ \nu \n< \n~ u.. \n\n0 \n\n.2 \n\no \n\no \n\n5 \n\n10 \n\nITERATION x 102 \n\nFigure 2: Plot of the fraction of neurons firing per unit time for different values of \nE: (a) 0.45, (b) 0.58, and (c) 0.72. At t = 0 all neurons except the central source are \nquiescent. At t = 500 (indicated by .J-) the source is shut off. The region indicated \nby (*) corresponds to an epoch in which spiral tip breakup occurs. \n\n\f1004 \n\nMilton, Chu, and Cowan \n\nThe temporal dynamics of the network can be examined by plotting the fraction F \nof neurons that fire as a function of time. As E is increased through target waves \n(Figure 2a) to spiral waves (Figure 2b) to disorganized patterns (Figure 2c), the \nfluctuations in F become less regular, the mean value increases and the amplitude \ndecreases. On closer inspection it can be seen that during spiral wave propagation \n(Figure 2b) the time series for F undergoes amplitude modulation as reported \npreviously (Farley, 1965). The interval of low amplitUde, very irregular fluctuations \nin F (* in Figure 2b) corresponds to a period of spiral tip breakup (Figure lc). \n\nThe appearance of spiral waves is typically preceded by 20-30 target waves. The \nformation of a spiral wave appears to occur in two steps. First there is an increase \nin the minimum value of F which begins at t '\" 420 and more abruptly occurs at \nt '\" 460 (Figure 2b). The target waves first become asymmetric and then activity \npropagates from the source region without the more centrally located neurons first \nentering the quisecent state (Figure 3c). At this time the spatially coherent wave \nfront of the target waves becomes replaced by a disordered noncoherent distribution \nof active and refractory neurons. Secondly, the dispersed network activity begins \nto coalesce (Figures 3c and 3d) until at t '\" 536 the first identifiable spiral occurs \n(Figure 3e). \n\nFigure 3: The fraction of neurons firing per unit time, for differing values of gener(cid:173)\nation time t: (a) 175, (b) 345, (c) 465, (d) 503 (e) 536, and (f) 749. At t = 0 all \nneurons except the central source are quiescent. \n\nIt was found that only 4 out of 20 networks constructed with the same 0, j3 produced \nspiral waves for E = 0.58 with periodic central point stimulation (simulations, in \nsome cases, ran up to 50,000 generations). However, for all 20 networks, spiral \nwaves could be obtained by the use of non-uniform initial conditions. Moreover, \nfor those networks in which spiral waves occurred, the generation at which they \nformed differed. These observations emphasize that small fluctuations in the local \nconnectivity of neurons likely play a major role in governing the dynamics of the \nnetwork. \n\n\fSpiral Waves in Integrate-and-Fire Neural Networks \n\n1005 \n\n4 Discussion \n\nSelf-maintaining spiral waves can a..rise in an inhomogeneous neural network with \nuniform initial conditions. Initially well-formed target waves emanate periodically \nfrom the centrally placed source. Eventually, provided that E is in a critical range \n(Figures 1 & 3), the target waves may break up and be replaced by spiral waves. \nThe necessary conditions for spiral wave formation are that: 1) the network be \nsufficiently tightly connected (Farley, 1965; Farley and Clark, 1961) and 2) the \nprobability of interneuronal connectivity should decrease with distance (unpublished \nobservations). As the network is made more tightly connected the probability that \nself-maintained activity arises increases provided that E is in the appropriate range \n(unpublished observations). These criteria are not sufficient to ensure that self(cid:173)\nmaintained activity, including spiral waves, will form in a given realization of the \nneural network. It has previously been shown that partially formed spiral-like waves \ncan arise from periodic point stimulation in a model excitable media in which the \ninhomogeneity arises from a dispersion of refractory times, k;l (Kaplan, et al, \n1988). \n\nIntegrate-and-fire neural networks have two stable states: a state in which all \nneurons are at rest, another associated with spiral waves. Target waves represent \na transient response to perturbations away from the stable rest state. Since the \nneurons have memory (i.e. \nk m ), \nthe mean threshold and membrane potential of the network evolve with time. As \na consequence the mean fraction of firing neurons slowly increases (Figure 2b). \nOur simulations suggest that at some point, provided that the connectivity of the \nnetwork is suitable, the rest state suddenly becomes unstable and is replaced by \na stable spiral wave. This exchange of stability is typical of a sub-critical Hopf \nbifurcation. \n\nthere is a relative refractory state with ko \u00ab \n\nAlthough complex, but organized, spatio-temporal patterns of spreading activity \ncan readily be generated by a randomly connected neural network, the significance \nof these phenomena, if any, is not presently clear. On the one hand it is not difficult \nto imagine that these spatio-temporal dynamics could be related to phenomena \nranging from the generation of the EEG, to the spread of epileptic and migraine \nrelated activity and the transmission of visual images in the cortex to the formation \nof patterns and learning by artificial neural networks. On the other hand, the oc(cid:173)\ncurence of such phenomena in artificial neural nets could conceivably hinder efficient \nlearning, for example, by slowing convergence. Continued study of the properties \nof these networks will clearly be necessary before these issues can be resolved. \n\nAcknowledgements \n\nThe authors acknowledge useful discussions with Drs. G. B. Ermentrout, L. Glass \nand D. Kaplan and financial support from the National Institutes of Health (JM), \nthe Brain Research Foundation (JDC, JM), and the Office of Naval Research (JDC). \n\nReferences \n\nR. L. Beurle. (1956) Properties of a mass of cells capable of regenerating pulses. \nPhil. Trans. Roy. Soc. Lond. 240 B, 55-94. \n\n\f1006 \n\nMilton, Chu, and Cowan \n\nR. L. BeUl-Ie. (1962) FUnctional organization in random networks. In Principles of \nSelf-Organization, H. v. Foerster and G. W. Zopf, eds., pp 291-314. New York, \nPergamon Press. \nB. G. Farley_ (1965) A neuronal network model and the \"slow potentials\" of elec(cid:173)\ntrophysiology. Compo in Biomed_ Res. 2, 265-294. \nB. G. Farley & W. A. Clark. (1961) Activity in networks of neuron-like elements. \nIn Information Theory, C. Cherry, ed., pp 242-251. Washington, Butterworths. \nD. T. Kaplan, J. M.Smith,B. E. H. Saxberg & R. J. Cohen. \ndynamics in cardiac conduction. Math. Biosci. 90, 19-48. \n\n(1988) Nonlinear \n\nA. T. Winfree. \nPrinceton, N.J. \n\n(1987) When Time Breaks Down, Princeton University Press, \n\n\f", "award": [], "sourceid": 689, "authors": [{"given_name": "John", "family_name": "Milton", "institution": null}, {"given_name": "Po", "family_name": "Chu", "institution": null}, {"given_name": "Jack", "family_name": "Cowan", "institution": null}]}