Spontaneous Neural Firing in Biological & Artificial Neural Systems

Christian R. Huyck and Richard Bowles

Middlesex University, London, UK,

email: C.Huyck@mdx.ac.uk, web site:
http://www.cwa.mdx.ac.uk/chris/chrisroot.html

September 21, 2001

Abstract Neurons spontaneously fire. This neurophysiological
phenomenon has computational ramifications and encourages useful short
and long term dynamics in the brain. The noise of spontaneous neural
activation may actually improve the system.

Spontaneous neural firing can attenuate neural waves. This allows the
waves to carry more information and to stay active for longer.

Spontaneous neural activation also improves Cell Assembly (CA)
dynamics. Neurons that receive little or no activation are unused. We
argue and describe an experiment that shows that spontaneous
activation stabilises unused neurons. Spontaneous activation, along
with a post-Hebbian learning rule, puts the synaptic weights of these
neurons in a neutral position. This may encourage later recruitment
into CAs.

Spontaneous neural activation also encourages recruitment of neurons
into nearby CAs. We describe an experiment that shows neurons are
recruited into nearby CAs when there is spontaneous activation, and
remain isolated with no spontaneous activation.

The neurophysiological phenomena of spontaneous neural firing has at
least three computational rationales. It encourages more information
travelling in neural waves. It allows unused neurons to remain stable
until they are eventually recruited into CAs. It encourages neural
recruitment into CAs.

1 Background and Introduction 
Neurons generally fire when they receive activation from other
neurons. However, there is evidence to show that they can also fire
spontaneously, i.e. without any external activation (Bevan and Wilson
, 1999; Abeles et al. , 1993). (Zeki , 1993) suggests that spontaneous
firing of neurons in the visual cortex may be responsible for
hallucinations. Neurons generally only fire spontaneously when they
have been inactive for a long period (relative to their normal firing
rate), typically several milliseconds. When a neuron has been active,
it fatigues, which tends to discourage it from firing until it has
recovered from that fatigue.

There is also some evidence that spontaneous activation is used to set
neuronal function. Spontaneous retinal discharges is necessary for the
segregation of the afferents from the two eyes in ocular dominance
columns (Singer , 1995; Stryker and Harris , 1986). Spontaneous
activation is necessary to set neurons in parts of early visual
processing. There is no reason to assume that it is not needed
elsewhere in the brain.

Spontaneous neural firing may have some physiological reason, but
neurons are computational devices. What are the computational
ramifications of spontaneous neural firing?

The overall physiological reason for spontaneous neural firing may
simply be, like the appendix, an accident of evolution. Still, it,
like flies' wings, requires some expenditure of energy. When flies are
bred in an environment that prevents them from using their wings,
flies with wings are bred out of the population. In the evolutionary
race, the expense of wings is too much when there is no
reward. Similarly, the expense of spontaneous neural firing might not
be justified, from an evolutionary perspective, if it did not have a
useful function. Computational research suggests two useful functions:
attenuation of neural waves, and recruitment of neurons to Cell
Assemblies(CAs).

2 Attenuation of Neural Waves 
Beurle (Beurle , 1956) models neural functioning as a wave of active
neurons. Imagine a pebble dropped in a pond with ripples spreading
from the centre. In the brain an area is activated (the pebble is
dropped), and activation spreads from that point (the ripples). The
original area ceases to be active due to neural fatigue, leaving a
wave of active neurons centred about the original area of
activation. This argument assumes two dimensions, but generalises to
three dimensions. Circular waves of activation are replaced by spheres
of activation.

The height of the ripple is analogous to the percentage of neurons
firing in the wave. If the height is too low, then the wave will die
out because new (as yet unactivated) neurons will not get enough
stimulus to fire.

When a wave is absent from an area of the cortex (i.e. none of the
neurons in that particular area is firing), there is clearly no
information content. Similarly, when all the neurons in that area are
firing there is very little information present, since there is no
differentiation of one subarea from any other. However, when only some
of the neurons are firing, there is the possibility of different
patterns of information being encoded, and that the level of
activation may increase or decrease. Therefore, an almost essential
requirement for information to be present in the patterns of neural
firing is that some of the neurons do not fire.

Assuming that the wave represents a concept, all neurons firing
corresponds to the perfect instance of the concept, the prototype
(Rosch and Mervis , 1975). Some neurons not firing corresponds to a
non-prototypical concept; these non-firing neurons carry information
about properties that are not present in the currently active
concept. For example a fully firing bird wave might refer to a typical
robin (considered to be the perfect example of a bird), while an
incomplete wave might refer to a penguin (a bird lacking the
archetypal quality of flight), or a dead robin.

Spontaneous neural firing makes it less likely that all neurons in the
area of the wave will fire. This is due to the dynamics of firing and
to the dynamics of learning.

The short-term dynamics of neural firing are affected by spontaneous
firing. When a neuron fires, spontaneously or not, it is less likely
to fire again due to fatigue. More importantly, spontaneous firing may
lead to other neurons firing; thus many neurons may be fatigued. These
fatigued neurons will be less likely to fire when the wave passes
through.

The dynamics of learning are also influenced by spontaneous
firing. Anti-Hebbian learning rules reduce the strength of connections
when one neuron connected to a synapse fires and the other does
not. Using anti-Hebbian learning rules (Shouval and Perrone , 1995),
spontaneously fired neurons may reduce the strength of already strong
connections. Thus activated neurons will be less likely to activate
other neurons, leading to a reduced height of the wave.

3 Recruitment of Neurons to a CA 
Buerle's wave model is related to Hebb's CA hypothesis. CAs are
reverberating circuits of neurons. The idea comes from Donald Hebb
(Hebb , 1949), who proposed the model as a mechanism for short and
long-term memory, for perception, for separating figure and ground,
and many other problems. A CA is a large number of neurons with many
connections between them. The activation within the loop is
self-sustaining. Once activated by external input, the CA as a whole
remains active longer than any individual neuron within it due to its
property of self-reactivation. The wave model relates to reverberating
cycles, but cycles may exist without waves. A CA is the neural
correlate of a concept.

3.1 Recruiting Nearby Neurons 
CAs form by a Hebbian learning mechanism. When two neurons fire
simultaneously, the connection between them, if it exists, is
strengthened. This local learning rule allows groups of neurons to
activate each other. These learned groups are CAs.

If a neuron is not part of a CA, it will only rarely, if ever, be
fired by activation from other neurons. This is because it has only
weak connections from other neurons. (If it had strong connections, it
would be part of a CA.) Since it has only weak connections, it never
gets enough activation to surpass the firing threshold. If a neuron
never fired it would be useless.

If, on the other hand, neuron A fires spontaneously, it may fire at
the same time as neuron B. Assume neuron B is connected to A and is in
a CA. When they both fire, the connection between the two neurons is
strengthened. If this happens often enough, the neuron becomes part of
the CA. Thus spontaneous firing of a neuron allows it to be recruited
into a CA.

3.2 Post-Hebbian Learning Rule 
In the developing brain, it may be possible that areas of the brain
receive little or no activation from other areas or from the
environment. Latter in development, these areas may become part of
CAs. This may relate to Piagetian stages (Inhelder and Piaget ,
1958). This lack of external activation may also occur with single
neurons that are initially poorly connected A mechanism that enabled
these weakly connected neurons to develop some connections without
external activation would make computational sense. There is a large
amount of cell death in infants. While this may be related to lack of
stimulus, it is very inefficient to kill off these neurons.

Initially it might appear that spontaneous activation would lead to
zero valued excitatory weights and inhibitory weights with values
tending to infinity in a portion of the network that does not receive
external stimulus. However, post-Hebbian learning rules keep the
weights at a neutral value. This neutral value might allow the neurons
to be easily recruited into CAs when external stimulus eventually
arrives.

When no CAs exist in an area and there is spontaneous activation,
naive learning rules can lead to all excitatory weights going to
zero. Anti-Hebbian learning rules are a model of long-term depression
(LTD). When one neuron fires and an adjacent neuron does not fire, the
weight of connecting synapses are reduced. LTD is used to prevent the
synaptic weights growing infinitely.

When neurons are far from CAs they will be stimulated mostly by
spontaneous activation. Since only a small percentage of neurons are
stimulated, it will be the rare case when adjacent neurons are
active. Thus, LTD will be applied much more frequently than long-term
potentiation (LTP).

There are two standard types of LTD pre-not-post and
post-not-pre. Both pre-not-post and post-not-pre LTD encourage weights
going to zero. Pre-not-post LTD occurs when the pre-synaptic neuron is
active and post-synaptic neuron is inactive. The spontaneously
activated neuron fires, and none of those it connects to fires; all of
its synaptic weights will be reduced and will tend toward
zero. Post-not-pre LTD occurs when the post-synaptic neuron is active
and the pre-synaptic neuron is inactive. The spontaneously activated
neuron fires and none of those neurons that feed it fires, so all the
incoming synapses have their weights reduced, again tending toward
zero.

The situation is even worse when inhibitory neurons are
concerned. Extending LTD to inhibitory neurons increases inhibition
when one neuron fires and the adjoining neuron does not. This means
that inhibition will increase without a limit.

Post-Hebbian learning rules (Shouval and Perrone , 1995) allow the
weight to avoid going to zero. In the standard Hebbian (LTP) and
Anti-Hebbian (LTD) learning rules, the change of weights is a
constant. Hebb proposes the rule \Delta w = jaiaj. In this equation j
is a constant and ai and aj are the activations of the neurons. In our
learning rule, learning only occurs when neurons fire, thus activation
levels can be considered as 1. In post-Hebbian learning rules, the
change may be based on the current synaptic strength. When the
strength is high the increase from LTP is small, but when it is low
the increase from LTP is large. Similarly, when strength is low, the
decrease from LTD is small and when the strength is high, the decrease
is large. In a network where the only stimulation comes from
spontaneous activation, all weights will tend toward a neutral value.

This argument is tested with a neural simulation. This simulation is
fully described in section 4.1.

4 Experiments 
We have implemented several simple CA models in software (for a
further description see (Huyck , 2000, 2001)). These models have a
small number of neurons (hundreds), but show how important spontaneous
neural firing is for recruiting neurons into CAs.

Our models are based on leaky integrator neurons. Each neuron has
activation, which it receives from other neurons or the
environment. If the neuron has enough activation to exceed the firing
threshold, it fires and sends activation down to its synapses. If the
neuron fires, it loses all of its activation. If the neuron does not
fire, it loses some but not all of its activation; the activation
leaks away.

Each neuron has many synapses and each synapse connects to another
neuron. Through learning, the strength of the synapse may change; this
models LTP and LTD. Activation is passed to the post-synaptic neuron
when the pre-synaptic neuron fires; the strength of the synapse is the
amount of activation that is passed. The neurons are connected in a
distance-biased fashion. That is, a neuron is more likely to be
connected to a nearby neuron than a distant one.

In this paper we describe two experiments. The first is running an
untrained network for many cycles and measuring the change in
connection strengths. This is a comparison between postHebbian
learning and standard Hebbian learning. As discussed in section 3.2,
this experiment shows that connection weights reach a reasonable
weight when post-Hebbian learning is used and unreasonable weights
when standard learning is used.

In the second experiment, two patterns are learned. When spontaneous
activation of neurons is added, neurons that are unaffiliated with a
pattern/CA are recruited into the CAs.

4.1 Experiment 1: Spontaneous Activation and Synaptic Weight 
In this experiment, the initially untrained network is allowed to run
for 1200 cycles. No input pattern is presented to the network, but
each neuron has a one percent chance of being spontaneously
activated. The network is run in two different modes. In the first
mode a post-Hebbian learning

Figure 1: Average Synaptic Strengths rule is used, and in the second mode a standard Hebbian learning rule is used. The post-Hebbian rule modifies the strength of the synapse depending on the existing strength of the synapse. The standard Hebbian rule modifies the strength by a constant.

The results of the test are shown in figure 1. The vertical axis
refers to total synaptic strength. The horizontal axis refers to the
number of cycles run. For both modes there are two lines; the first is
the sum of all the excitatory weights, and the second is the sum of
all the inhibitory weights. For both modes, the initial weights are
the same. These initial weights are somewhat arbitrary and the same
results should occur for most initial weight settings.

In the post-Hebbian case the weights of the excitatory neurons stay
stable from the start and the inhibitory weights converge around cycle
1200. In the Hebbian case, the excitatory neurons' weights get near to
zero within 500 cycles, and the inhibitory weights decrease without
limit.

With post-Hebbian learning, excitatory weights decrease due to
LTD. However, as the weights are already small, the decrease is
small. Occasionally, the neurons will be spontaneously activated
together and the strength will increase due to LTP. This increase will
be large because the synaptic strength is small. This interplay
between LTP and LTD allows the synapse to go to a neutral weight.

This experiment verifies the argument proposed in section
3.2. Spontaneous activation is a problem for learning based on
constant weights, but is not a problem for post-Hebbian learning
rules.

4.2 Experiment 2: Spontaneous Activation and CA Spread 
In this experiment, neural patterns are presented to the initial
net. Varying amounts of spontaneous activation allows unaffiliated
neurons to be recruited into CAs.

In this experiment, there is a 200 by 200 grid of neurons. There are two different patterns:
Figure 2: Neuron Column Activation Rate the first pattern is a band of
neurons between columns 30 and 70; the second pattern is a band of
neurons between columns 130 and 170. These patterns are modelled by
directly activating neurons. In a given run, only some of the neurons
in the pattern are activated externally. After repeated presentation,
the net has learned the patterns. It will classify a pattern by
activating the entire CA. Almost all neurons are activated via
internal reverberation once the CA has formed.

Figure 2 shows the frequency of firing of the cells in a cross section
through the rectangular grid of cells. The solid line represents
average neural activation when there is no spontaneous activation. The
key point is that there is no activation outside of the pattern. For
measurement purposes, spontaneous activation was turned off for
testing. The dotted line shows the behaviour when 5 percent of the
neurons are spontaneously activated during training. This shows that
many neurons outside the pattern are being activated, and thus have
been recruited into the CA. The dashed line shows a further increase
when 10 percent are spontaneously activated.

This experiment indicates that CAs do in fact recruit cells at their
edges which receive no activation other than spontaneous
activation. Clearly, increasing the probability of spontaneous
activation, Pr(SA), encourages recruitment, as the cell assemblies
start to spread into the territory of uncommitted cells. However, as
Pr(SA) is increased beyond about 0.3, the cell assemblies have
extended so far that cells tend to be recruited into both at the same
time. This results in a join between the cell assembly for pattern A
and that for pattern B, so that they effectively fuse into one large
CA. Increasing the spacing between the patterns reduces the likelihood
of this happening.

4.3 Experiment 3: CA Ignition from Spontaneous Activation 
It is perfectly reasonable that spontaneous activation alone should be
capable of igniting a CA. Often, when walking down the street, ideas
`pop into one's head' totally unbidden, and it is quite possible that
this occurs as the result of the spontaneous activation of a CA.

Figure 3: Time steps required for CA ignition A further experiment was
therefore designed to discover whether a previously-trained CA could
be activated solely by the presence of spontaneous activity occurring
in cells forming part of the CA. A grid of 20-by-20 cells each with a
connectivity of 6 to others within its immediate neighbourhood was
trained as follows: Cells within the central 10-by-10 square of the
grid each had a 20being activated at each time step. It was found that
no more than 50 time steps were required for a stable CA to form. No
spontaneous activation of the cells outside the central square was
permitted during training, although these cells could, of course, be
stimulated through their connections.

After training, the net was run taking spontaneous activation of cells
as its only input; all the cells had a specified spontaneous
activation probability at each time step. Experiments were repeated
100 times.

The number of time steps before the CA ignited was noted, ignition
being taken as the point when the activity level within the region
where the CA had previously formed rose above 70activity level in the
CA directly after training. Figure 3 shows the average number of time
steps for CA ignition.

The CA did not ignite in all the test runs, and the proportion of test
runs where ignition did occur was noted for all spontaneous activation
probabilities. Figure 4 shows likelihood of CA ignition.

Repeated experiments showe d that the length of time required before CA
ignition and the probability of that ignition in the first place
depended strongly on the spontaneous activation probability of the
cells. Unsurprisingly, the greater the level of spontaneous
activation, the greater the proportion of trials runs in which CA
ignition occurred. When the probability rose above 0.01, ignition of
the CA was virtually guaranteed.

5 Discussion and Conclusion 
Neurons spontaneously fire. This has positive ramifications for a
theory of neural processing.

Spontaneous firing should lead to more information content in neural
waves. This is due to the Figure 4: Frequency of spontaneous CA
ignition dynamics of firing and of learning being influenced by
spontaneous firing. Both of these processes make it less likely that
all of the neurons in a neural wave will not fire. Fewer neurons
firing increases information content.

Spontaneous firing also improves the dynamics of neural recruitment in
CAs. Our experiments show that spontaneous firing can lead to more
robust CAs.

The first experiment shows that spontaneous activation stabilises the
synaptic weights of areas of neurons that receive no external
stimulation. This may support colonisation of these areas by new CAs.

The second experiment shows that spontaneous activation is essential
to recruiting neurons into nearby CAs. Neurons near to CAs are easily
recruited when there is no spontaneous activation.

The third experiment showed that spontaneous activity could ignite
CAs. This indicates that concepts may simply become active
spontaneously. In a larger system this spontaneous ignition would
often need to be suppressed, perhaps by inhibition from active CAs.

Long-term CA dynamics will include CA formation, neural recruitment,
and CA fractionation. Concepts form, evolve, and break into
sub-concepts. Though it is not the sole reason for these operations,
spontaneous neural activation may facilitate all of them.

CAs by their very nature are dynamic. The number of neurons in a CA
will increase and decrease. Neurons may only have partial membership
in a CA. One of the cornerstones of CA theory is that neurons may
participate in multiple CAs (Sakurai , 1998). Future work will be
needed to explore these dynamics, but they will almost certainly be
influenced by spontaneous neural activation.

This work is by no means conclusive. It does give a computational
rationale for the neurophysiological phenomena of spontaneous neural
activation. It shows how spontaneous firing might lead to better
performance in real neural systems.

This work provides extra evidence for the CA hypothesis, and provides
an example of how computational modelling with information theoretic
goals can inform neurophysiology and neuropsychology. While we are a
long way from complete understanding of how neurons generate
behaviour, this work is another contribution toward that
understanding.

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