An oscillatory neural network model of sparse distributed
memory and novelty detection
Roman Borisyuk
a,c,*, Mike Denham
a, Frank Hoppensteadt
b,
Yakov Kazanovich
c, Olga Vinogradova
daCentre for Neural and Adapti6e Systems,School of Computing,Uni6ersity of Plymouth,Drake Circus,Plymouth PL4 8AA, UK bCenter for Systems Science,Arizona State Uni6ersity,Tempe,AZ85287-7606,USA
cInstitute of Mathematical Problems in Biology,Russian Academy of Sciences,Pushchino142290,Russia dInstitute of Theoretical and Experimental Biophysics,Russian Academy of Sciences,Pushchino142290,Russia
Abstract
A model of sparse distributed memory is developed that is based on phase relations between the incoming signals and an oscillatory mechanism for information processing. This includes phase-frequency encoding of input informa-tion, natural frequency adaptation among the network oscillators for storage of input signals, and a resonance amplification mechanism that responds to familiar stimuli. Simulations of this model show different types of dynamics in response to new and familiar stimuli. The application of the model to hippocampal working memory is discussed. © 2000 Elsevier Science Ireland Ltd. All rights reserved.
Keywords:Synchronisation; Memory formation; Frequency adaptation
www.elsevier.com/locate/biosystems
1. Introduction
Current opinion about information processing in the brain presumes that biological memory systems constantly make decisions concerning the storage of incoming information. One important attribute for deciding whether information should be stored or not is its novelty or significance to the organism. Novelty detection can be thought of as a differential response of some parts of the brain to a stimulus depending upon relations
be-tween the incoming and previously stored infor-mation. In this paper we develop an oscillatory neural network model that is able to store infor-mation and to respond with various types of dynamical behaviour to stimulation by new or familiar stimuli.
Recently Borisyuk and Hoppensteadt (1998) Borisyuk and Hoppensteadt (1999) developed an oscillator network model of memory in the hippocampus that is based on phase relations between one-dimensional periodic input signals supplied by two channels coming from the en-torhinal cortex and the medial septum, respec-tively. Depending on these relations, a particular part of the network is activated by a stimulus, and * Corresponding author. Tel.:+44-1752-232541; fax:+
44-1752-232540.
E-mail address:[email protected] (R. Borisyuk).
the memory is created through Hebbian-like mod-ification of connections between oscillators.
In developing our model here, we also use two input channels to the network. It is assumed that the first channel carries information about fea-tures of the stimulus, and the second channel delivers oscillatory activity to the network. The important new feature of the model is that a multidimensional periodic input signal supplied by the first channel is assumed. The components of the signal have the same frequency but differ by phase shifts. For a particular stimulus, mem-ory storage takes place in those regions of the network where there is a proper phase coincidence of the input signal components. Due to random time delays in connections constituting the first channel, a proper coincidence of phases of input signal components takes place at only a small number of network locations (in comparison to the size of the whole network). This results in a sparse distribution of high activity in the network during each stimulation and hence in sparse cod-ing of stimuli in the network memory. In addition the following new ideas appear in the model.
Memory storage of the input by appropriate
modification of natural frequencies of oscillators.
Memory retrieval by resonance of network
os-cillatory activity in response to an external input.
The idea of learning and memorising by form-ing an ensemble of frequency-tuned oscillators is traditional in the field of oscillatory neural net-works. Whilst in connectionist theory it is as-sumed that an ensemble of synchronous (in-phase) oscillators is formed via strengthening connections between oscillators in the ensemble, we suggest an alternative mechanism of memori-sation through adaptation of natural frequencies of oscillators. We suppose that input signals syn-chronise the activity of some oscillators in the network, resulting in a ‘learned’ pattern; that is, some oscillator frequencies are gradually tuned by network dynamics to the frequency of the input signals. Eventually, a population of recruited os-cillators, now having identical dynamical proper-ties, is formed.
The hypothesis that adaptation of oscillation frequencies can be used as a learning mechanism appeared in the works of Ukhtomsky and his school (Ukhtomsky 1978) and John (Thatcher and John 1977). Neural network implementation of this mechanism has been suggested by Torras (1986) and Hoppensteadt (1992).
The other component of our model that is used for choosing locations for memory storage and for retrieving previously learned information is frequency resonance between the input and some network oscillators. Suppose that oscillatory sig-nals of the frequency vare channelled in parallel with the different time delays to a set of oscilla-tors working with a variety of natural frequencies. Those oscillators that receive the input signals approximately in-phase and whose natural fre-quencies are near v can be most easily phase-locked by the input. The permanent coincidence of the oscillator phase with the phases of the input signals results in a sharp increase of tion amplitude while amplitudes of other oscilla-tors decay (on average).
2. Model description
Our mathematical model of novelty detection is a one-layer network of interacting oscillators (Fig. 1). The activity of an oscillator represents the average activity of interactive neural populations (local field potential). We suppose that oscillator dynamics might be described in terms of a phase-locking procedure. Such networks have been
ful in mathematical neuroscience when a qualita-tive mathematical representation of synchronisa-tion is needed (Ermentrout and Kopell, 1994; Kazanovich and Borisyuk, 1994, 1999; Hoppen-steadt, 1997; Hoppensteadt and Izhikevich, 1997). In our model, an oscillator is described by three variables: phase, natural frequency, and ampli-tude (see Appendix A). Such an oscillator can be considered as a generalisation of a phase oscillator.
Oscillators are combined into groups Gj (j=
1, … ,m) with q oscillators in each group. The oscillators belonging to the same group are cou-pled by all-to-all connections. For simplicity, there are no connections between groups.
There are two input channels that deliver the information about a stimulus to each oscillator in the network. The first channel supplies an n -di-mensional signal C=(C1, …, Cn), the second
channel supplies a one-dimensional signal S. The components of Care periodic oscillations, Ci=csin (v0t+cij). The amplitude c and the
frequencyv0are the same for all the components.
The amplitude c is binary (1 during stimulation and 0 otherwise). The frequency v0 depends on
the stimulus. Each stimulus is associated with a particular value of v0 taken from some range
(vmin,vmax). Besides the frequency, the oscillatory
signal in the first channel is coded by a set of phase shifts cij (i=1, …, n, j=1, …, m), where cij represents the oscillation phase shift of the
component Ci at the inputs of oscillators in the
group Gj. These phase shifts imitate different
time-lags during input signal transmission to a particular group of oscillators. We suppose that phase shifts cij are randomly distributed in the
range (0, t).
The signal S is equal to v0 during stimulation
and 0 otherwise. During stimulation, the oscilla-tors of the network start to work at their natural frequencies generating low amplitude oscillations. Further evolution of an oscillator’s activity during stimulation depends mostly on the signalC. After stimulation is over (S=0) the activity in the network decays to zero.
In the initial state (before the oscillatory net-work stores any information) each groupGj
con-tains oscillators whose natural frequencies are
distributed in the whole range (vmin, vmax) of
input frequencies. During information storage these natural frequencies may change.
A basic assumption of our modelling is that an oscillator reaches and keeps a high level of activ-ity if the signals that are supplied to this oscillator through the first channel arrive in-phase, that is if the values of cij are approximately the same for
the givenj. Due to a random choice of the values of cij for each stimulus, this implies that the
presentation of a stimulus results in a high oscilla-tory activity at only a small number of randomly chosen locations (groups), where an appropriate coincidence of input signal phases takes place. The activity in other parts of the network is low. This type of activity appears during both memori-sation and recall. We call it a sparse representa-tion (coding) of stimuli in network activity. The important feature of sparse coding is that if the number of groups in the network is large relative to the number of memorised stimuli, then differ-ent stimuli (even those that are labelled by the same or similar frequencies) will activate different (though possibly overlapping) regions in the net-work. Therefore, different stimuli will be memo-rised in different locations of the network.
In this paper we are mostly interested in de-scribing how this sparse representation is imple-mented, therefore, we will consider stimuli that are coded by the same frequencyv0and different
sets of phase shifts. The memorisation of stimuli coded by different frequencies and approximately the same values of phase shifts has been consid-ered in (Borisyuk et al., 2000).
Memory storage is conditioned by dynamics of the oscillators organised in a special way and related adaptation of their natural frequencies. Here we describe it in a non-formal way, focusing on the algorithms embedded in the network.
on the synchrony of its oscillations with the sig-nals from the first channel. This causes a selective amplification of the activity of some oscillators.
The following two principles were applied to control the activity of an oscillator.
Principle 1. The amplitude of oscillations sharply increases if most of the signals of the first channel arrive at an oscillator in-phase with its own oscillations. In particular, this implies that this oscillator is synchronised with the input and operates with the current frequency v0. We
con-sider this increase as a resonant response of an oscillator to a properly tuned input signal. The amplitudes of those oscillators that do not work in-phase with C are kept at a low level or decrease.
It is important to note that resonance is a necessary condition for the effective synchronising influence of an oscillator on the other oscillators of its group. The connection strength between coupled oscillators is made too low to synchronise their activity in a non-resonant state. In particu-lar, in the absence of the signalCno synchronisa-tion of oscillasynchronisa-tions in the network is possible. In a resonant state an oscillator is capable of synchro-nising the activity of those oscillators of its group whose natural frequencies are near the frequency of the resonant oscillation.
Principle 2. The natural frequency of an oscilla-tor tends to the value of its current frequency. To avoid rapid jumps of natural frequencies during transitional stages of synchronisation, the dynam-ics of the natural frequencies are made slow rela-tive to the rate of phase locking.
Let us describe how these principles are embed-ded in the control of the network dynamics.
Suppose that a stimulus is presented at the input of the network. According to Principle 1, under the influence of the signalC, some oscilla-tors of the network increase their activity and reach a resonant state. The amplitudes of other oscillators are kept low.
Oscillators in a resonant state play the role of a source for spreading synchronisation to other os-cillators of the group to which they belong. This is the basic mechanism for memory storage in the network; a stimulus is coded in the network mem-ory by a sufficiently large population of oscillators
with natural frequencies nearly identical to that of the input. Such a population is formed during and immediately after stimulation as a result of two processes. First, some oscillators in the group (with natural frequencies nearv0) are recruited to
synchronisation and resonance by oscillators that are already resonant. Second, due to Principle 2 the synchronisation results in these oscillators ‘learning’ the new frequency. Thus these natural frequencies are gradually tuned to the frequency
v0.
After several oscillators in a group reach a resonant state, they combine their efforts to re-cruit other oscillators of the group to synchronisa-tion and resonance. The array of natural frequencies that have been formed during stimula-tion is conserved and later these frequencies are used as initial natural frequencies of oscillators during presentations of other stimuli. Note that the memory in the form of adapted natural fre-quencies is of a static type in the sense that it is supposed to be unchanged in the periods between stimuli presentations when the network is silent.
As the learning of a given stimulus continues in a session of repeated stimulations, a sufficiently large population of oscillators in a resonant state (with nearly identical natural frequencies) ap-pears. The size of the population of oscillators in a resonant state at the end of a current stimula-tion is used as an indicator to distinguish between new and familiar stimuli. By computer simulation we show that it is possible to choose the parame-ters of learning control in such a way that for a new stimulus the number of resonant oscillators at the end of stimulus presentation is small but it becomes large (and exceeds a certain threshold) if a stimulus has been learnt before.
3. The simulation
For simulation we used a network with the number of groups m=100 and the number of oscillators in the groups q=50. The dimension of the signal in the first channel was n=20.
coded by the same frequencyv0=7 and a set of
time delays {cij s} (
s=1,2,3) randomly distributed in the range (0, p/2). During the learning proce-dure each stimulus was presented twice in succes-sion, at the times (2s−2)T and (2s−1)T (s=1,2,3). The duration of each stimulation was T=100. Before learning, the natural frequencies
vkof the oscillators in each group of the network
were distributed with a fixed step in the range (6.5, 7.5) so that vk+1−vk=const.
The parameters of the network were chosen in such a way that one presentation of a stimulus was sufficient to memorise it, so the network reacted to further presentations of this stimulus as to a familiar stimulus. Each stimulus was coded by resonant activity in about 10 – 15 groups (out of 100).
Stimulation was accompanied by modification of oscillator’s natural frequencies and amplitudes. The natural frequencies of the oscillators ob-tained during an interval of stimulation of length T were used as the initial values of the natural frequencies for the next stimulus presentation. These modified natural frequencies were the only information stored about the stimuli. The ampli-tudes and phases were assigned zero values at the beginning of each stimulation (amplitude and phase reset).
The results of simulations are presented in Figs. 2 – 4. Fig. 2 shows the dynamics of the natural frequencies of the oscillators in two groups. Fig. 2a shows a group where resonant activity appeared in response to stimulation by the first stimulus but there was no resonant re-sponse to stimulation by the second stimulus. Analogously, Fig. 2b shows a group sensitive to stimulation by the second stimulus but insensitive to stimulation by the first stimulus. It can be seen that adaptation of the natural frequencies takes place in the case of a resonant response only. The evolution of the natural frequencies of those os-cillators that were close tov0 started earlier and
was faster. After the same stimulus has been presented for the second time, all the oscillators of the corresponding group achieved approxi-mately the same natural frequencies.
Fig. 3 shows the dynamics of the amplitudes in the same groups. Those oscillators that have been
Fig. 2. Time evolution of the natural frequencies in two groups of oscillators responding by resonant activity to stimulation by the first (a) and second (b) stimuli, respectively. The range of initial values of the natural frequencies is (6.5, 7.5). The stimuli are coded by the frequencyv0=7. The duration of a
stimula-tion is 100. Note that stimulastimula-tion results in the adaptastimula-tion of the natural frequencies to the frequency of the input signal in appropriate groups of oscillators.
stimulus all oscillators of the corresponding group were in a resonant state.
To test the reliability of novelty detection by the network, we fixed two parameters. The first parameter was used to detect a resonance state. An oscillator was considered as being in a reso-nant state if its amplitude was higher than 2/3 of the amplitude’s maximum value. The second parameter was used to detect novelty. A stimulus was considered as unknown if at the end of stimulation the number of oscillators in a reso-nant state was less than a threshold value R= 400. Fig. 4 shows the dynamics of the number of oscillators N in a resonant state. It can be seen
that the variableNis below the threshold during Fig. 4. Time evolution of the number of oscillators in a resonant state. The number of different stimuli is 3, each stimulus being repeated twice in succession. The threshold for discrimination between new and familiar stimuli is 400.
Fig. 3. Time evolution of the amplitudes of oscillations in the first (a) and second (b) groups of oscillators introduced in Fig. 2. Note the increasing number of oscillators involved in a resonant state during repeated presentations of a stimulus.
the first presentation of a stimulus but crosses the threshold during the second presentation of each stimulus.
4. Discussion
Our model development has been inspired by evidence of memory storage and novelty detection in the hippocampus. It is interesting to compare anatomical and neurophysiological data about the role of the septo-hippocampal system in these phenomena with the results of our modelling.
Physiologi-cal data suggest that the signals in both inputs to the hippocampus contain theta-rhythm frequen-cies in the range 4 – 9 Hz (Vinogradova 1995; Iijima et al. 1996). The spatial distribution of inputs from the entorhinal cortex cause time de-lays, which result in phase lags which may be as high as one half of period of the theta-rhythm (Miller, 1991).
It is hypothesised that the hippocampus plays a co-ordinating role in forming the information flow in the brain (Damasio, 1989) and is used as temporary information storage for the time inter-val between information coding and permanent storage in the neocortex (Squire, 1992). The the-ory of cortico-hippocampal interplay (Miller 1991) postulates that the memory is formed by the appropriate resonant loops between the entorhi-nal cortex and the hippocampus, each with spe-cific propagation time.
There is evidently a correspondence between the septo-cortico-hippocampal system and com-ponents of the model described above. The net-work represents the hippocampus, a group of oscillators is supposed to belong to a hippocam-pal slice. The input C is the signal from the entorhinal cortex modulated by the theta-rhythm, Sis the signal from the medial septum that elicits theta-rhythm activity in the hippocampus.
In the model, we ignore the backward loop from the hippocampus to the cortex. This loop is known to be used for hippocampal control of information processing in the cortex and for long-term memory storage. On the contrary, the mem-ory in the hippocampus that we model is of a temporary type (working memory). In terms of the model, we assume that oscillator tunings are not kept for a long time. Without constant repeti-tion of the same stimulus the oscillators that code this stimulus gradually ‘recall’ their original natu-ral frequencies. Thus, memory about this particu-lar stimulus is freed for the storage of other stimuli.
Also, we do not model inhibition loops to suppress hippocampal theta-activity and stop memory storage in the case of a familiar stimulus. A simple model of such an inhibitory mechanism is presented in (Borisyuk et al., 2000).
Acknowledgements
The work of R. Borisyuk, Y. Kazanovich, and O. Vinogradova was supported in part by the Russian Foundation of Basic Research (Grant 99 – 04 – 49112 for R. Borisyuk and Y. Ka-zanovich, Grant 99 – 04 – 48281 for O. Vinogra-dova). F. Hoppensteadt was supported in part by NSF Grant 98 – 05544. Y. Kazanovich was sup-ported by a University of Plymouth Visiting Fel-lowship, funded by Invensys plc.
Appendix A
The network consists of m(m=100) groups of oscillators. Each group containsq(q=50) oscilla-tors. The dynamics of the network is governed by the following equations for the oscillator phase
uk
j, amplitudea k
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