56O 
A MODEL OF NEURAL OSCILITOR FOR A UNIFIED Sb]MOIXLE 
A. B. Kirillov, G.N. Borisyuk, R.M. Borisyuk, 
Ye. I. Kovalenko, V.I. Makarenko, V. A. Chulaevsky, 
V. I. Kryukov 
Research Computer Center 
USSR Academy of Sciences 
Pushchino, Moscow Region 
142292 USSR 
A new model of a controlled neuron oscillator, 
proposed earlier {Kryukov et al, 1986} for the 
interpretation of the neural activity in various 
parts of the central nervous system, may have 
important applications in eineering and in the 
theory of brain functions. The oscillator has a 
good stability of the oscillation period, its 
frequency is regulated linearly in a wide range 
and it can exhibit arbitrarily long oscillation 
periods without changing the time constants of 
its elements. The latter is achieved by using 
the critical slowdown in the dynamics arising in 
a network of nonformal excitatory neurons 
{Kovalenko et al, 1984, Kryukov, 1984}. By 
changing the parameters of the oscillator one 
can obtain various functional modes which are 
necessary to develop a model of higher brain 
function. 
Or oscillator comprises several hundreds of modelled 
excitatory neurons (located at the sites of a plane lattice) 
and one inhibitory neuron. The latter receives output 
signals from all the excitatory neurons and its own output 
is transmitted via fccdback to every excitatory neuron (Fig. 
1). Each excitmtory neuron is connected bilaterally with its 
four nearest neighhours. 
Each neuron has a threshold r(t) decaying exponentially to a 
A Model of Neural Oscillator for a Unified Submodule 561 
value r e or r (for an excitatory or inhibitory neuron). A 
Gaussian noise with zero mean and standard deviation ( is 
added to a threshold. A membrane potential of a neuron is 
the sum of input impulses decaying exponentially when there 
are no input. If the membrane potential excccds the 
threshold, the neuron fires and sends impulses to the 
neighbouring neurons. An impulse from excitatory neuron to 
excitatory one increases the membrane potential of the 
latter by aee, from the excitatory to the inhibitory - by 
aei, and from the inhibitory to the excitatory - decreases 
the membrane potential by aie. We consider a discrete time 
model, the time step being equal to the absolute refractory 
neuron. 
We associate a variable xi(t) with each excitatory 
If the l-th neuron fires at step t, we take xi(t)=l; 
if it 
does not, then x i(t)=0. The mean g(t)=l/N x i(t) will be 
referred to as the network activity, where N is the number 
of excitatory neurons. 
A 
Figure 1. A- neuron, B- scheme of interconnections 
Let us consider a situation when inhibitory fcdback is cut 
off. Then such a model exhibits a critical slowdown of the 
dynamics {Kovalenko et al, 1984, Kryukov, 1984}. Namely, if 
the interconnections and parameters of neurons are chosen 
appropriately, initial pattern of activated neurons has an 
unusually long lifetime as compared with the time of membrane 
potential decay. In this mode g(t) is slowly increasing and 
562 Kirillov, et al 
causes the inhibitory neuron to fire. 
Now, if we urn on the negative fccdback, output impulse 
from inhibitory neuron sharply decreases membrane potentials 
of excitatory neurons. As a a consequence, E(t) falls down 
and process starts from the beginning. 
We studied this oscillator by means of 
There are 400 excitatory neurons (20*20 
inhibitory neuron in our model. 
simulation model. 
lattice) and one 
a. When the thresholds of excitatory neurons are high 
enough, the inhibitory neuron does not fire and there are no 
A 
Tcp- 49 + 3 t 
............... 6 ..... /b ' 
B 
11 I I I I I I I I I I [ I I 1 
I.! I  I  I1 I I I I ! I [I 11 t z 
[ I I I I I ! I I I I I I II 
Ill 
_. 1_ I __ ______j 
[11 Itl 11 !llll II I 'rLa 
Figure 2. Oscillatory mode. A - network activity, 
B- neuron spike trains 
oscillations. 
b. At lower values of r the network activity E(t) changes 
periodically and excitatory neurons generate bursts of 
spikes (Fig. 2). The inhibitory neuron generates regular 
periodical spike trains. 
c. If the parameters are chosen appropriately, the mean 
oscillation period is mch greater than the mean interspike 
interval of a network neuron. The frequency of oscillations 
is regulated by r (Fig. 3A) or, which is the same, by the 
A Model of Neural Oscillator for a Unified Submodule 563 
intensity of the input flow. The min'nnum period is 
determined by the decay rate of the inhibi*ry input, the 
maximum - by the lifetime of the metastable state. 
6O 
5O 
4O 
3O 
o 
IO 
A B 
9. I0. II. 12. 13. 7,, 
4C  ILC I,'C LCC :Z 
Fu 3. A - oscillation frequency 1/T vs. threshold r e, 
B - coefficient of variation of the period K vs. period 
d. The coefficient of variation of the period is of the 
order of several percent, but it increases at low 
frequencies (Fig. 3B). The stability of oscillations can be 
increased by introducing some inhomogeneity in the network, 
for example, when a part of excitatory neurons will receive 
no inhibitory signals. 
CCILITO UNDER  STATION 
In this section we consider first the neural network without 
the inhibitory neuron. But we imitate a periodic input to 
the network by slowly varying the thresholds r(t) of the 
excitatory neurons. Namely, we add to r(t) a value 
Ar=-A, sin(lt) and fire a part of the network at some phase of 
the sine wave. Then we look at the time nccded for the 
network to restore its background activity. There are 
specific values of a phase for which this time is rather big 
(Fig. 4A). Now consider the full ocsillator with an 
oscillation period T (in this section T=-85+_2.5 time steps). 
We stimulate the oscillator by periodical (with 
tst<35) sharp increase of membrane potential 
excitatory neuron by a value ast. As the 
procccds, the oscillation period 
T--5 to some value Tst, remaining 
the period 
of each 
stimulation 
gradually decreases from 
then equal to Tst. The 
564 Kirillov, et al 
value of Tst depends on the stimulation intensity ast: 
gets greater, Tst tends to the stimulation period tst. 
as a t 
A 0 
 '" B 
10 
0 
10 20 tst 
Figure 4. A - threshold modulation, B- duration of the 
network responce vs. phase of threshold modulation, 
- critical stimulation intensity vs. stimulation period 
For every stimulation period tst there is cb2racteristic 
value a 0 of the stimulation intensity ast, such that with 
ast>aO the value of Tst is equal to the stimulation period 
tst. The dependence between a 0 and tst is close to a linear 
one (Fig. 4B). The usual relaxation oscillator also exibits 
a linear dependence between a and tst. At the same time, we 
did not find in our oscillator any resonance phenomena 
essential to a linear oscillator. 
In a further development of the neural oscillator we tried 
to build a model that will be more adequate to the 
biological counterpart. To this end, we changed the 
structure of interconnections and tried to define more 
correctly the noise component of the input signal coming to 
an excitatory neuron. In the model described above we 
A Model of Neural Oscillator for a Unified Submodule 565 
imitated the sum of inputs from distant neurons by 
independent Gaussian noise. Here we used real noise produced 
by the network. 
In order to simulate this internal noise, we randomly choose 
16 distant neighbours for every exitatory neuron . Then we 
assume that the network elements are adjusted to work in a 
certain noise environment. This means that a 'mean' internal 
noise would provide conditions for the neuron to be the most 
sensitive for the information coming from its nearest 
neighbors. 
So, for every neuron i we calculate the sum ki=xj(t), where 
summation is over all distant neighbors of this neuron, and 
compare it with the mean internal noise k=l/N Rk.. The 
internal noise for the neuron i now is ni--C(ki-k), where 
is a constant. 
We choose model parameters in such a way that the noise 
component is of the order of several percent of the membrane 
potential. Nevertheless, the network exhibits in this case a 
dramatic increase of the lifetime of initial pattern of 
activated neurons, as compared with the network with 
independent Gaussian noise. A range of parameters, for which 
this slowdown of the dynamics is observed, is also 
considerably irireased. Hence, longer periods and better 
period stability could be obtained for our generator if we 
use internal noise. 
THE CHAIN OF THREE SUBMODULES: A MODEL OF COLUMN OSCILLATOR 
Now we consider a small system constituted of three 
oscillator submodules, A, B and C, connected consecutively 
so that submodule A can transmit excitation to submodule B, 
B to C, and C to A. The excitation can only be transmitted 
when the total activity of the submodule reaches its 
threshold level, i.e. when the corresponding inhibitory 
neuron fires. After the inhibitory neuron has fired, the 
activity of its submodule is set to be small enough for the 
submodule not to be active with large probability until the 
excitation from another submodule comes. Therefore, we 
expect A, B and C to work consecutively. In fact, in our 
simulation experiments we observed such behavior of the 
566 Kirillov, et al 
S(T) 
2O 
A T 
........... 15 
10 12 
10 12 
Figure5. Chain of thrcc submodules. Period of 
oscillations (A) and its standard deviation (B) vs. 
noise amplitude 
closed chain of 3 basic submodules. The activity of the 
whole system is nearly periodic. Figure 5A displays the 
period T vs. the noise amplitude O. The scale of O is chosen 
so that 0.5 corresponds approximately to the resting 
potential. An interesting feature of the chain is that the 
standard deviation $(T) of the period (Fig. 5B) is small 
enough, even for the oscillator of relatively small size. 
The upper lines in Fig. 5 correspond to square 10.10 
network, middle - to 9*9, lower - to 8*8 one. One can scc 
that the loss of 36 percent of elements only causes a 
reduction of the working range without the loss of 
stability. 
OONCLUSION 
Though we have not considered all the interesting modes of 
the oscillator, we believe that, owing to the phenomenon of 
metastability, the same oscillator exhibits different 
behaviour under slightly different threshold parameters and 
the same and/or different inputs. 
Let us enumerate the most 
possibilities of the oscillator, 
obtained from our results. 
interesting functional 
which can be easily 
1.Pacemaker with the frequency regulated in a wide range and 
with a high period stability, as compared with the neuron 
(Fig. 3B). 
2. Integrator (input=threshold, output=phase) with a wide 
A Model of Neural Oscillator for a Unified Submodule 567 
range of linear regulation (see Fig. 3A). 
3.Generator of damped oscillations (for discontinuous input). 
4.Delay device controlled by an external signal. 
5. Phase comparator (see Fig. 4A). 
We have already used these functions for the interpretation 
of electrical activity of several functionally different 
neural structures {Kryukov et al, 1986}. The other functions 
will be used in a system model of attention {Kryukov, 1989} 
presented in this volume. All these considerations justify 
the name of our neural oscillator - a unified submodule for 
a 'resonance' neurocomter. 
I. Kovalenko, G. N. Borisyuk, R. M. Borisyuk, A. B. 
Kirillov,V. I. Kryukov. Short-term memory as a 
roetastable state. II.Simulation model, ybernetics and 
Systems Research, 2, R. Trappl (ed.), Elsevier, pp. 
266-270 (1984) 
I. Kryukov. Short-term memory 
I .Master equation approach, 
Reseaz-, 2, R. Trappl (ed.), 
(1984) 
as a roetastable state. 
yberne t i cs and Systems 
Elsevier, pp. 261-265 
V. I. Kryukov. "Neurolocator", a model of attention 
(1989)(in this volume). 
V. I. Kryukov, G. N. Borisyuk, R. M. Borisyuk, A. B. 
Kirillov, Ye. I. Kovalenko. The Metastable and Unstable 
States in the Brain (in Russian), Pushchino, Acad. Sci. 
USSR (1986) (to appear in Stochtio Cellular Systems: 
Ergodicity, Memory, Morphogenesis, Manchester University 
Press, 1989). 
