How Oscillatory Neuronal Responses Reflect 
Bistability and Switching of the Hidden 
Assembly Dynamics 
K. Pawelzik, H.-U. Bauer t, J. Deppisch, and T. Geisel 
Institut fiir Theoretische Physik and SFB 185 Nichtlineare Dynamik 
Universitat Frankfurt, Robert-Mayer-Str. 8-10, D-6000 Frankfurt/M. 11, FRG 
ttemporary adress:CNS-Program, Caltech 216-76, Pasadena 
emMl: klaus@chaos.uni-frankfurt.dbp.de 
Abstract 
A switching between apparently coherent (oscillatory) and stochastic 
episodes of activity has been observed in responses from cat and monkey 
visual cortex. We describe the dynamics of these phenomena in two paral- 
lel approaches, a phenomenological and a rather microscopic one. On the 
one hand we analyze neuronal responses in terms of a hidden state model 
(HSM). The parameters of this model are extracted directly from exper- 
imental spike trains. They characterize the underlying dynamics as well 
as the coupling of individual neurons to the network. This phenomenolog- 
ical model thus provides a new framework for the experimental analysis 
of network dynamics. The application of this method to multi unit ac- 
tivities from the visual cortex of the cat substantiates the existence of 
oscillatory and stochastic states and quantifies the switching behaviour 
in the assembly dynamics. On the other hand we start from the single 
spiking neuron and derive a master equation for the time evolution of the 
assembly state which we represent by a phase density. This phase density 
dynamics (PDD) exhibits costability of two attractors, a limit cycle, and 
a fixed point when synaptic interaction is nonlinear. External fluctuations 
can switch the bistable system from one state to the other. Finally we 
show, that the two approaches are mutually consistent and therefore both 
explain the detailed time structure in the data. 
977 
978 Pawelzik, Bauer, Deppisch, and Geisel 
I INTRODUCTION 
A few years ago, oscillatory and synchronous neuronal activity was discovered in 
cat visual cortex [1-3]. These experiments backed earlier considerations about syn- 
chrony in neuronal activity as a mechanism to bind features, e.g., of an object in a 
visual scene [4]. They triggered broad experimental and theoretical investigations 
of detailed neuronal dynamics as a means for information processing and, in par- 
ticular, for feature binding. Many theoretical contributions tried to reproduce and 
explain aspects of the experimentally observed phenomena [5]. Motivated by the 
experiments, the models where particularly designed to exhibit spatial synchroniza- 
tion of permanent oscillatory responses upon stimulation by a common, connected 
stimulus like a bar. Most models consist of elements which exhibit a limit cycle 
after a simple Hopf bifurcation. 
The experimental data, however, contain many details which the present models do 
not yet completely incorporate. One of these details is the coexistence of regular 
and irregular episodes in the data, which interchange in an apparently stochastic 
manner. This interchange can be observed in the signals from a single electrode [6] 
as well as in the time-resolved correlation of the signals from two electrodes [7]. In 
this contribution we show, that the observed time structure reflects a switching in 
the dynamics of the underlying neuronal system. This will be demonstrated by two 
complementary approaches: 
On the one hand we present a new method for a quantitative analysis of the dynam- 
ical system underlying the measured spike trains. Our approach gives a quantitative 
description of the dynamical phenomena and furthermore explains the relation be- 
tween the collective excitation in the network which is not accessible experimentally 
(i.e. hidden) and the contributions of the single observed neurons in terms of transi- 
tion probability functions. These probabilities are the parameters of our Ansatz and 
can be estimated directly from multi unit activities (MUA) using the Baum-Welch- 
algorithm. Especially for the data from cat visual cortex we find that indeed there 
are two states dominating the dynamics of collective excitation, namely a state of 
repeated excitation and a state in which the observed neurons fire independently 
and stochastically. 
On the other hand using simple statistical considerations we derive a description 
for a local neuronal subpopulation which exhibits bistability. The dynamics of 
the subpopulation can either rest on a fixed point - corresponding to the irregular 
firing patterns - or can follow a limit cycle - corresponding to the oscillatory firing 
patterns. The subpopulation can alternate between both states under the influence 
of noise in the external excitation. It turns out that the dynamics of this formal 
model reproduces the observed local cortical signals in much detail. 
2 Excitability of Neurons and Neuronal Assemblies 
An abstract model of a neuron under external excitation e is given by its threshold 
dynamics. The state of the neuron is represented by its phase s, which is the 
time passed by since the last action potential (s = 0). The threshold t3 is high 
directly after a spike and falls off in time and the neuron can fire again when e 
exceeds t3. In case of noise or internal stochasticity, an excitability description of 
Oscillatory Neuronal Responses Reflect Bistability & Switching of Assembly Dynamics 979 
the dynamics of the neuron is more adequate. It gives the probability PI to fire 
again in dependence of the state q5  with Pl(q5 ) = or(e- (qSs)) and cr some sigmoid 
function. A monotonously falling threshold  then corresponds to a monotonously 
increasing excitability PI' Such a description neglects any memory in the neuron 
going beyond the last spike. In particular this means for an isolated neuron, that 
PI can be easily calculated from the inter-spike interval histogram (ISIH) Ph using 
the relation Ph(t) = Pl(t). (1 - fJ Pn(tt)dt'). In that case also the autocorrelation 
function can be calculated from Pn(t) via ((r) = Pn(r) + f Pn(r)((r- t)dt. 
The excitability formulation sketched above is not valid for a neuron which is em- 
bedded in a neuronal assembly. However, we may use this Ansatz of a renewal 
process to describe the activation dynamics of the whole assembly (see section 5). 
The phase q5 b = 0 here corresponds to the state of synchronous activity of many 
neurons in the assembly, which we call burst for convenience. Since the dynamics 
of the network can differ from the dynamics of the elements we expect the func- 
tion P}(q5 b) which now describes the burst excitability of the whole assembly to be 
different from,the spike excitability Pl(qSs) of the single neuron. 
A simple example for this is a system of integrate and fire neurons in which os- 
ciliatory and irregular phases emerge under fixed stimulus conditions([8, 9] and 
section 5). Contrary to the excitability of the single refractory element the burst 
excitability P) of the system has a maximum at q5 b = T which expresses the in- 
creased probability to burst again after the typical oscillation period T, i.e. the 
maximum represents a state o of oscillation. The assembly, however, can miss to 
burst around q5  = T with a probability Po--, and switch into a second state s 
in which the probability P,--,o to burst again is reduced to a constant level. The 
switching probabilities Po--, and P,--,o can be easily calculated from P). In this 
way the shape of P) distinguishes a system with an oscillatory state from a system 
which is purely refractory but which nevertheless can still have strong modulations 
in the autocorrelogram [13]. 
3 Hidden states and stochastic observables 
The single neuron in an assembly, however, need not be strictly coupled to the state 
of the assembly, i.e. a neuron may also spike for qb > 0 and it may not take part 
in a burst. This stochastic coupling to an underlying process suffices to destroy 
the equivalence of Ph and the autocorrelogram C(r) =< s(t)s(t + r) >t of the 
spike train s(t)  {0, 1} (Fig 1). We therefore include the probability Po,(c) ) to 
observe a spike when the assembly is in the state q5  into our description (Fig. 2). 
The unlikely case where the spike represents the burst corresponds to the choice 
Pob = 6b,o. 
4 Application to Experimental Data 
While our approach is quite general, we here concentrate on the measurements 
of Gray et al. [2] in cat visual cortex. Because our hidden state model has the 
structure of a hidden Markov model we can obtain all the parameters Pood(c)) and 
980 Pawelzik, Bauer, Deppisch, and Geisel 
0.08 
0.06 
0.04 
0.02 
0.00 I , , , I , , , I   , 
0 20 40 60 80 
Figure 1: Correlogram of multi unit activities from cat visual cortex (line). Corre- 
lax)grams predicted from the ISIH (/k) and from the hidden state model (4-). 
Measurement Level [ Spike Event 
/1- p.(O) / 1- p(1) / 
Network Level / 'f'' 1J f / .... I 'f' ' / pf(n 1- 
pf(n) 
Figure 2: The hidden state model. While P)(b b) governs the dynamics of assembly 
states c) b, Pob, (q 5) represents the probability to observe a spike of a single neuron. 
Oscillatory Neuronal Responses Reflect Bistability & Switching of Assembly Dynamics 981 
Figure 3: Network excitability P) and single neuron contribution Pob, estimated 
from experimental spike trains (A17, cat). 
P)(b) directly from the multi unit activities using the well known Baum-Welch 
algorithm[10]. The results can be seen in Fig. 3. The excitability shows a typical 
peak around the main period at T = 19ms, which indicates a state of oscillation. 
For larger phases we see a reduced excitability which reveals a state of stochastic 
activity (Pob,(q5  > T) > 0). The spike observation probability Po,(q5) is peaked 
near the burst and is about constant elsewhere. This means that we can characterize 
the data by a stochastic switching between two dynamical states in the underlying 
system. Because of the stochastic coupling of the single neuron to the assembly state 
this can only hardly be observed directly. The switching probabilities between either 
states calculated from P) coincide with results from other methods [11]. 
From the excitability P) and the spike probabilities Poo, we now obtain the autocor- 
relation function (r) = f f, Po,(c)OM(c),c))rPob(c))p(c))dc)dc), with M being 
the transition matrix of the Markov model (see also below). The result is compared 
to the true autocorrelation C(r) in Fig. 1. The excellent agreement confirms our 
simple Ansatz of a renewal process for the hidden burst dynamics of the assembly. 
5 Bistability and Switching in Networks of Spiking Neurons 
The above results indicate that the dynamics of a cortical assembly includes bista- 
bility rather than a simple Hopf bifurcation. In order to understand how this bista- 
bility emerges in a network we go one step back and derive a model for a neuronal 
subpopulation on the basis of spiking neurons. We assume again that the internal 
state of the neuron is given by the threshold function 13 depending on the time since 
the last spike event and that the excitability of the neuron can be described by a 
982 Pawelzik, Bauer, Deppisch, and Geisel 
x 
-o 
c 
time 
Figure 4: Illustration of assembly state representation by a phase density. 
firing probability Pf. In a network, however, the input to the neuron has external 
contributions iet as well as from within iint i.e. 
Pf(65,/) -- sigm(weztiezt(l) -t'- wintiint(l) -- 
For a more formal treatment of the dynamics of such a network we characterize the 
assembly state by a phase density (65, t) which gives the relative amount of neurons 
in the assembly which are in phase 65 at time t (Fig 4). 
Discretizing the internal phases 65, we transform k(65, t) to /(j), a vector whose 
components i give the probability to be in phase 65i 6 [(i-1)At,jAr] at time 
tj = jar. The number T of components is chosen large enough to ensure that 
pl(T,j) = P(TAt,jAt) does not change any more. This vector evolves in time 
according to 
/(j + 1) = M(j)/Y(j) 
(1) 
with 
( pi (T, j) h 
M(j) - 
0 pi(1,j) pf(2,j) ... pi(T-l,j) 
1 0 0 
0 1-pf(1,j) 0 ... 0 
1-p (2, j) ". 
1-pi(T,j) 
= g(Yo). 
o 
o ... 1-pi(T-l,j) 
beeing a matrix that incorporates the effects of firing (reset) via the firing probability 
pf(i,j). 
It remains to define the lateral interaction in the subpopulation. Clearly only the 
fraction of neurons that fire cem interact, therefore we have 
Oscillatory Neuronal Responses Reflect Bistability & Switching of Assembly Dynamics 983 
1.0 
0.8 
06 
0.4 
0.2 
0.0 
0 
200 400 600 800 1000 
t 
Figure 5' Switching in the assembly dynamics under external fluctuations. 
that P0 denotes the time dependent firing rate. 
Note 
Numerically iterating the dynamics (1) we find, that the distribution can evolve in 
two distinct ways, depending on the lateral interaction function g and the initial- 
ization. First fi can relax to a fixed point, corresponding to a constant fraction of 
the neurons firing at a particular time. On the level of an individual neuron, this 
corresponds to a stochastic firing characteristics, and in the measurements this state 
corresponds to the irregular periods. Secondly the distribution can evolve according 
to a limit cycle, with a rather large fraction of the neurons in a narrow band of phases 
i.e. the neurons are synchronous. We find parameter combinations wext, wint where 
both states coexist, i.e. we find bistability, if the interaction function is nonlinear, 
like g(x) cr x  or g(x) cr x 4. Nonlinearities of this kind can be brought about by 
a different type of preferred synapses with a nonlinear transmission characteristic 
(like an NMDA-receptor synapse) for the corticocortical projections, as compared 
to the thalamocortical projections. At this point we only want to make the point 
that our simple model system can exhibit bistability under reasonable assumptions. 
In the bistable regime some initializations of fi lead to the oscillatory state, some 
to the stochastic state. Adding some noise to the external input the system can 
also switch between the two states(Fig. 5). In this way our simple local model can 
capture the switching phenomenon which is inherent in the experimental data [12]. 
6 Summary and Synthesis 
We presented two complementary approaches to the dynamics of neuronal subpopu- 
lations, a phenomenological one which captures the time structure in measured spike 
trains and a neuronal one which provides a formal description of the dynamics of 
assemblies of spiking neurons. In the phenomenological approach we introduced 
the hidden state model which revealed that the system underlying the multi unit 
activities from the cat switches between states of oscillatory and stochastic activity. 
The analysis of the phase density dynamics showed, that bistability and switching 
emerges in networks of spiking neurons when the neuronal interaction is nonlinear. 
It remains to show that these approaches are also quantitatively consistent, i.e. that 
they are two sides of the same medal. Instead of a formal proof we only remark here 
that the parameters of the HSM can be extracted directly from the dynamics of the 
PDD under external noise. For this purpose one only needs to evaluate the inter- 
burst interval distribution from which P) can be calculated. Pob, is easily estimated 
as the average shape of po(t) between successive bursts. We find, that already this 
procedure gives HSMs which accurately reproduce the correlation function of the 
984 Pawelzik, Bauer, Deppisch, and Geisel 
full firing dynamics po(t). This means that the HSM captures relevant aspects of 
the assembly dynamics including the relation between the network dynamics and 
the contributions of single neurons. 
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