Spiral Waves in Integrate-and-Fire 
Neural Networks 
John G. Milton 
Department of Neurology 
The University of Chicago 
Chicago, IL 60637 
Po Hsiang Chu 
Department of Computer Science 
DcPaul University 
Chicago, IL 60614 
Jack D. Cowan 
Department of Mathematics 
The University of Chicago 
Chicago, IL 60637 
Abstract 
The formation of propagating spiral waves is studied in a randomly 
connected neural network composed of integrate-and-fire neurons 
with recovery period and cxcitatory connections using computer 
simulations. Network activity is initiated by periodic stimulation 
at a single point. The results suggest that spiral waves can arise in 
such a network via a sub-critical Hopf bifurcation. 
1 Introduction 
In neural networks activity propagates through populations, or layers, of neurons. 
This propagation can be monitored as an evolution of spatial patterns of activity. 
Thirty years ago, computer simulations on the spread of activity through 2-D ran- 
domly connected networks demonstrated that a variety of complex spatio-tcmporal 
patterns can be generated including target waves and spirals (Bcurlc, 1956, 1962; 
Farley and Clark, 1961; Farley, 1965). The networks studied by these investigators 
correspond to inhomogeneous excitable media in which the probability of intcrncu- 
ronal connectivity decreases exponentially with distance. Although travelling spiral 
waves can readily be formed in excitable media by the introduction of non-uniform 
1001 
1002 Milton, Chu, and Cowan 
initial conditions (e.g. Winfree, 1987), this approach is not suitable for the study 
and classification of the dynamics associated with the onset of spiral wave forma- 
tion. Here we show that spiral waves can "spontaneously" arise from target waves 
in a neural network in which activity is initiated by periodic stimulation at a sin- 
gle point. In particular, the onset of spiral wave formation appears to occur via a 
sub-critical Hopf bifurcation. 
2 Methods 
Computer simulations were used to simulate the propagation of activity from a 
centrally placed source in a neural network containing 100 x 100 neurons arranged 
on a square lattice with excitatory interactions. At t = 0 all neurons were at rest 
except the source. There were free boundary conditions and all simulations were 
performed on a SUN SPARC 1+ computer. 
The network was constructed by assuming that the probability, A, of intcrncuronal 
connectivity was an exponential decreasing function of distance, i.e. 
 =/3 exp(-clr]) 
where c = 0.6,/3 = 1.5 are constants and Irl is the euclidean interneuronal distance 
(on average each neuron makes 24 connections and ~ 1.3 connections per neuron, 
i.e. multiple connections occur). Once the connectivity was determined it remained 
fixed throughout the simulation. 
The dynamics of each neuron were represented by an integrate-and-fire model pos- 
sessing a "leaky" membrane potential and an absolute (1 time step) and relative 
refractory or recovery period as described previously (Bcurlc, 1962; Farley, 1965; 
Farley and Clark, 1961): the membrane and threshold decay constants were, respec- 
tively, k, = 0.3 mscc -1, ko = 0.03 mscc -1. The time step of the network was taken 
as I mscc and it was assumed that during this time a neuron transmits excitation 
to all other neurons connected to it. 
3 Results 
We illustrate the dynamics of a particular network as a function of the magnitude 
of the cxcitatory intcrncuronal excitation, E, when all other parameters are fixed. 
When E < 0.2 no activity propagates from the central source. For 0.2 _< E < 0.58 
target waves regularly emanate from the centrally placed source (Figure la). For 
E _> 0.58 the activity patterns, once established, persisted even when the source 
was turned off. Complex spiral waves occurred when 0.58 _< E < 0.63 (Figures 
lb-ld). In these cases spiral meandering, spiral tip break-up and the formation of 
new spirals (some with multiple arms) occur continuously. Eventually the spirals 
tend to migrate out of the network. For E _> 0.63 only disorganized spatial patterns 
occurred without clearly distinguishable wave fronts, except initially (Figures 1e-f). 
Spiral Waves in Integrate-and-Fire Neural Networks 1003 
Figure 1: Representative examples of the spatial pattern of neural activity as a 
function of E:(a) E = 0.45, (b - c) E = 0.58 and (f) E = 0.72. Color code: gray = 
quiescent, white = activated, black = relatively refractory. See text for details. 
5 
 .14 
u.. 
z 
o 
,, .07 
z 
z 
_o o 
t- 
J .2 
u.. 
b I 
ITERATION x 10 2' 
Figure 2: Plot of the fraction of neurons firing per unit tinhe for different values of 
E: (a) 0.45, (b) 0.58, and (c) 0.72. At t = 0 all neurons except the central source arc 
quiescent. At t = 500 (indicated by $) the source is shut off. The region indicated 
by (,) corresponds to an epoch in which spiral tip breakup occurs. 
1004 Milton, Chu, and Cowan 
The temporal dynamics of the network can be examined by plotting the fraction F 
of neurons that fire as a function of time. As E is increased through target waves 
(Figure 2a) to spiral waves (Figure 2b) to disorganized patterns (Figure 2c), the 
fluctuations in F become less regular, the mean value increases and the amplitude 
decreases. On closer inspection it can be seen that during spiral wave propagation 
(Figure 2b) the time series for F undergoes amplitude modulation as reported 
previously (Farlcy 1965). The interval of low amplitude, very irregular fluctuations 
in F (* in Figure 2b) corresponds to a period of spiral tip breakup (Figure 1c). 
The appearance of spiral waves is typically preceded by 20-30 target waves. The 
formation of a spiral wave appears to occur in two steps. First there is an increase 
in the minimum value of F which begins at t  420 and more abruptly occurs at 
t ~ 460 (Figure 2b). The target waves first become asymmetric and then activity 
propagates from the source region without the more centrally located neurons first 
entering the quiscccnt state (Figure 3c). At this time the spatially coherent wave 
front of the target waves becomes replaced by a disordered noncoherent distribution 
of active and refractory neurons. Secondly, the dispersed network activity begins 
to coalesce (Figures 3c and 3d) until at t ,, 536 the first identifiable spiral occurs 
(Figure 3c). 
Figure 3: The fraction of neurons firing per unit time, for differing values of gener- 
ation time t: (a) 175, (b) 345, (c) 465, (d) 503 (c) 536, and (f) 749. At t = 0 all 
neurons except the central source are quiescent. 
It was found that only 4 out of 20 networks constructed with the same a,/ produced 
spiral waves for E = 0.58 with periodic central point stimulation (simulations, in 
some cases, ran up to 50,000 generations). However, for all 20 networks, spiral 
waves could be obtained by the use of non-uniform initial conditions. Moreover, 
for those networks in which spiral waves occurred, the generation at which they 
formed differed. These observations emphasize that small fluctuations in the local 
connectivity of neurons likely play a major role in governing the dynamics of the 
network. 
Spiral Waves in Integrate-and-Fire Neural Networks 1005 
4 Discussion 
Self-maintaining spiral waves can acise in an inhomogeneous neural network with 
uniform initial conditions. Initially well-formed target waves emanate periodically 
from the centrally placed source. Eventually, provided that E is in a critical range 
(Figures I & 3), the target waves may break up and be replaced by spiral waves. 
The necessary conditions for spiral wave formation are that: 1) the network be 
sufficiently tightly connected (Farley, 1965; Farley and Clark, 1961) and 2) the 
probability of intcrncuronal connectivity should decrease with distance (unpublished 
observations). As the network is made more tightly connected the probability that 
self-maintained activity arises increases provided that E is in the appropriate range 
(unpublished observations). These criteria are not sufficient to ensure that self- 
maintained activity, including spiral waves, will form in a given realization of the 
neural network. It has previously been shown that partially formed spiral-like waves 
can arise from periodic point stimulation in a model excitable media in which the 
inhomogeneity arises from a dispersion of refractory times, k -1 (Kaplan, ct al, 
1088). 
Integrate-and-fire neural networks have two stable states: a state in which all 
neurons are at rest, another associated with spiral waves. Target waves represent 
a transient response to perturbations away from the stable rest state. Since the 
neurons have memory (i.e. there is a relative refractory state with ko << kin), 
the mean threshold and membrane potential of the network evolve with time. As 
a consequence the mean fraction of firing neurons slowly increases (Figure 2b). 
Our simulations suggest that at some point, provided that the connectivity of the 
network is suitable, the rest state suddenly becomes unstable and is replaced by 
a stable spiral wave. This exchange of stability is typical of a sub-critical Hopf 
bifurcation. 
Although complex, but organized, spatio-temporal patterns of spreading activity 
can readily be generated by a randomly connected neural network, the significance 
of these phenomena, if any, is not presently clear. On the one hand it is not difficult 
to imagine that these spatio-tcmporal dynamics could be related to phenomena 
ranging from the generation of the EEG, to the spread of epileptic and migraine 
related activity and the transmission of visual images in the cortex to the formation 
of patterns and learning by artificial neural networks. On the other hand, the oc- 
curonce of such phenomena in artificial neural nets could conceivably hinder efficient 
learning, for example, by slowing convergence. Continued study of the properties 
of these networks will clearly be necessary before these issues can be resolved. 
Acknowledge ments 
The authors acknowledge useful discussions with Drs. G. B. Ermentrout, L. Glass 
and D. Kaplan and financial support from the National Institutes of Health (JM), 
the Brain Research Foundation (JDC, JM), and the Office of Naval Research (JDC). 
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