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```SECTION 7
The Accumulator Network
In Section 2 experimental data was discussed suggesting that an accumulator
representation of elapsed time might have the following properties: a) a
quantitative value, b) a mean that increases approximately linearly and c)
scalar variance. An abstract model of the accumulator neural network was
designed to produce all of these properties. The model is a modification of the
accumulator model proposed by Miall (1993) which has some plausibility as a
result of the observations of Niki and Watanabe (1979). In Miall’s model
single nodes become activated and stay active for the course of the interval
delay. In this model nodes do not necessary stay active for the course of the
interval delay, rather activation spreads around the network, each node
becoming active for only a short temporal unit unless re-excited. In short, the
accumulator neural network is an abstract model that simulates the
preservation of activation probabilities. The following description elaborates
the design considerations and implementation of the system.
7.1 Design and Implementation
Each node in the accumulator network has a fixed quantity of connections
(Nc) with other nodes that it can excite.
This connectivity was randomly
generated. Each node can be active or inactive. An active node can take on
different activation levels. The activation level of a node determines its
probability of activating other nodes. The activation level is the sum of
discrete activation units (Av) a node possesses. Each Av unit is equal to the
value of one. A node acquires a single Av unit for each successful input it
45
receives. A node is active when it receives one or more Av units. On each
time step the network receives a fixed quantity of inputs (Ni) from an external
source which can be viewed as the pulse generator in SET.
Each of the
inputs transmits a single Av unit to a node in the network, making that node
active. Every node activated by an external source is selected at random,
although this is not of any relevance to the performance of the algorithm.
The dynamics of the network unfolds over a sequence of idealised time steps.
In this model, nodes are active for a single time step unless they receive
activation on the next time step.
Each active node has a probability of
activating other nodes via its connection’s.
This activation transfer occurs
over each time step. The algorithm does not include intermediate states
between neurons receiving input and firing, refractory periods or any other
neural properties that unfold in real time.
One constraint is that the mean sum of Av units in the neural network (Sa) for
each time step should increase approximately linearly over the sequence of
time steps. For this to occur, a node should have a probability of activating
on average one other node for each Av unit it has on that time step. The
probability of an active node x activating another node y on a subsequent time
step is calculated by dividing the activation level of node x (sum of Av units)
by the value of Nc. If for example Nc equals 10 and the node x has an
activation level of 2 Av units, then each output node (y1…..y10) will have a
probability of being activated equal to 0.2.
46
To determine whether an input node x activates an output node y on the
subsequent time step the following rule is computed:
If 1/Nc  Rn then y=y+1
(7.1)
where Rn is a random number between 0 and 1 drawn from a uniform
distribution of such values. On each time step the algorithm searches for an
active node in the network. When one is found the rule in 7.1 is computed for
each of the nodes it can excite via its connections. If the input node x has an
activation value greater than 1, then the rule in 7.1 is computed once for each
Av unit it has. When all of the activation calculations have been performed
for a single active node x, the algorithm moves on to the next active node.
This process is continued for each active node in the neural network. When
this is completed, a new time step is initiated and the pattern of activation
transfer calculated from the previous time step, together with the new external
inputs, becomes the present activation pattern of the neural network.
The neural network should tend to preserve the quantity of activation it
receives. The mean sum of activation units in the neural network on a given
time step (t) can be predicted by multiplying the value of Ni by t. A second
constraint relates to maintaining scalar variance.
It was assumed that if
transfer of activation involved constant probabilities, then noise in the system
would increase proportionally with the value of Sa. A sample of Sa at t should
be a Gaussian distribution with some standard deviation (SD) that increases
monotonically with the size of t. The CV fraction depends of the size of Nc
47
and Ni. The greater Nc or lower Ni, the larger the value of CV. The author is
not clear of the precise mathematical relations between Nc, Ni and CV.
7.2 Results
The data reported shows 100 simulations of 100 time steps. Ni was set at 10
and Nc was set at 5. These simulations provide a sample (n=100) of Sa for
each time step. As can be seen from figure 7.1 the mean value of Sa increases
in a linear fashion as a function of time steps. The SD of sample of Sa values
on each time step also increases linearly as shown in figure 7.2. These two
observations predict that CV must approximate a constant fraction.
This is
shown in figure 7.3 where for each time step, the SD of the sample of Sa
values is divided by the mean Sa value and plotted as a function of time steps.
A last prediction was that a sample of Sa values at t should be a Gaussian
distribution. This is shown in figure 7.4 where the value Sa at eighty time
steps was taken from each of 100 simulations. The histogram of data shows
that the distribution approximates a Gaussian.
Figures 7.1-7.4: Data from accumulator network
Figure 7.1
Figure 7.2
1200
Standard deviation of the sum of activation units
250
Sum of activation units
1000
800
600
400
200
0
0
10
20
30
40
50
60
Time steps
70
80
90
200
150
100
50
0
100
48
0
10
20
30
40
50
60
Time steps
70
80
90
100
Figure 7.4
Figure 7.3
20
0.2
15
0.15
Frequency
Coeficient of variation of sum of activation units
0.25
0.1
5
0.05
0
10
0
10
20
30
40
50
60
Time steps
70
80
90
0
100
317 361 405 449 493 537 581 625 669 713
Sum of activation units
Each figure is computed from 100 simulations of 100 time steps (Nc=5, Ni=10). Figure 1 shows the mean Sa
as a function of time steps. Figure 2 shows the SD of a sample of Sa as a function of time steps. Figure 3
shows the CV fraction as function of time steps. Figure 4 show a histogram of a sample of Sa at eighty time
steps.
7.3 Discussion
As mentioned this model was designed to simulate activation probabilities
although it can also be conceptualised in more concrete terms.
Each Av unit
possessed by an active node, on each time step, can be thought of a single
active neuron. The number of neurons, like the sum of activation units,
increases over time steps in a linear fashion. Activation spreads around the
neural network although the number of active neurons in the network evoked
by an external input tends to preserve.
The neural network provides data in line with inferences based on empirical
evidence as discussed in Section 2.
This model is however very abstract.
The main reason for these simulations was to establish whether a network of
nodes with accumulating activity could generate the CV fraction through
49
noisy interactions.
The model shows that if activity increases in a
probabilistic fashion and the amount of probable noise is proportional to the
activity then scalar variance will emerge.
The pacemaker is modelled as a constant input to the neural network.
Speeding up or slowing down the internal clock can be manipulated by
increasing or decreasing the amount of Ni on each time step. The effect of
information load on the subjective passage of time (e.g. Ornstein, 1969) could
also be related to the amount of input in this model. This may occur as a
result of the amount of neural activity reflecting the degree of information
processed.
As we have discussed each Av unit can be thought of as a single active
neuron.
Lets assume the that the accumulator has a limited number of
neurons and whatever mechanisms modulate the activation in the neural
network, blindly activates already active neurons. As neurons do not have an
infinite firing rate, and it is assumed that the firing rate of single neurons
determines their probability of activating other neurons; one would expect a
fraction of the representation of elapsed time to be reduced. This reduction
would increase the longer time elapses, as more neurons become active. This
is an important issue relating to conceptualising the plausibility of this kind of
model.
Should one assume that single neurons always receive an equal
amount of excitation or should one assume that some neurons receive
disproportionately large amounts of excitation, as more neurons in the
network become active. For the former one is required to posit the view that
either a very sophisticated regulation mechanism is being employed or the
network consists of an extremely large number of neurons, much more than
50
will ever be simultaneously activated. For the latter one needs to assume the
position that elapsed time estimates reduce to some fraction (depending on the
network size and Ni) as a function of the interval length.
Noise in the system relates directly to the number of weights through which a
node can output. This relation between noise and weights was an unintended
consequence of developing an algorithm that would preserve all activity in the
neural network. As this generated noise, no further need was considered to
put noise anywhere else in the system. One important question concerns what
might cause this scalar noise to occur in a real neural network? One might
assume that if the number of active processors (neurons) represents elapsed
time, then something that is noisy and associated with the activity of single
processors would give rise to the scalar variance.
This could be a number of
things including response thresholds, synapses or some unknown mechanism
that modulates the activity.
Exactly how hypothetical accumulator neurons could give rise to, or be
modulated by external mechanisms to produce linearity and scalar variance is
not an easy question even attempt to answer. One speculative possibility is
that some pattern of connectivity and/or dynamical time dependent properties
of hypothetical accumulator neurons permit these properties to occur.
Another possibility is that the properties are accomplished by some kind of
feedback mechanism that through neurochemical mediation modulates
relevant time dependent neural properties. Either way this is just imprecise
speculation. The central issue in making an activation based timing
accumulator seem plausible is developing mechanisms that could do this.
51
Neither Miall’s (1993) model or this model includes such a mechanism. This
should be the central goal for any further developments.
The last issue concerns how the neural network could be reset.
One
possibility is that the input to the accumulator may be associated with the
presence or absence of a representation evoked by the reward situation. A
working memory of an expected reward event consisting of a population of
firing neurons could maintain sustained input to a neural network acting as a
pacemaker or indirectly influencing this input.
How might the activity be
reset? The simplest idea would be some form of inhibition that inputs to the
neural network and cancels out the excitation.
52
53
```
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