Appendix 2. How the TraceLink
paradigm works.
Introduction
TraceLink is
a connectionist paradigm introduced in the papers of Murre and Meeter on the
TraceLink model of amnesia (Murre, 1996; Murre & Meeter, subm.). The last
section, from Murre and Meeter (subm.) and for afficionados only, explains in
detail the working of this paradigm. The first two sections focus on how
TraceLink is implemented in Nutshell, and gives only a cursory explanation of
the paradigm itself.
The paradigm
uses Hebbian learning, and a simple linear integration of inputs. The two
distinguishing features of the model are the fact that nodes fire
stochastically, and the soft kWinnerTakeAll
character of layers in TraceLink networks.
The
likelihood with which a node fires is a function of the balance of inhibition
and excitation: if a node receives more excitation than inhibition, the
likelihood is greater than 0.5, and it is smaller than 0.5 if inhibition
dominates excitation. Excitation is the sum of inputs from the other nodes in
the network, inhibition is a variable, equal for all nodes in a layer, whose
composition will be explained later.
The degree of randomness is controlled by the 'temperature' parameter.
If the temperature is high (e.g., 1), then the nodes fire in a very random way,
and the difference between excitation and inhibition must be large to have an
impact on the node. If the temperature is low (e.g., 0.1), small differences
between excitation and inhibition are enough to result in a very high, or
alternatively very low, likelihood of firing.
TraceLink
implements a soft kWinnerTakeAll firing
regime. Inhibition is regulated in such a way that the fraction of nodes that
fires in a given layer is kept as close as possible to the fraction k given by the user. If a fraction
bigger than k of the nodes fire, then
inhibition is raised so that less nodes will be able to fire in the next
iteration. If a fraction smaller than k
of the nodes fires, then inhibition is lowered so that more nodes will be able
to fire. This process will continue until the fraction of firing nodes reaches
a stable state of around k (this process of settling
can take between 20 and 200 iterations).
Inhibition is
the sum of two amounts: fast inhibition (T) and slow inhibition (tau). Fast
inhibition is adjusted, as a name suggests, more rapidly than slow inhibition.
Both, however, are adjusted with the objective of bringing the fraction of
nodes that is active close to k.
Parameters of the
TraceLink paradigm in Nutshell
On the node level
A node has
four parameters: 'Act', 'Net', 'Clamped' and 'Deactivated'. 'Act' stands for
'Activation', and can have the values 1 (active, fires) or 0 (inactive). 'Net'
is the sum of excitatory inputs that the node received in the last iteration. A
node is 'Clamped' if its activity state is not updated, i.e. the node stays in
a its activity state independent of the input it might receive. A 'Deactivated'
node is lesioned; it does not function anymore.
On the tract and connection level
A connection
between two nodes has only one parameter: its 'weight'. This parameter has the
standard connectionist interpretation. Tracts are objects that contain all the
connections from all the nodes in one layer to all the nodes in another layer
(or all the other nodes in the same layer). They have five parameters. 'MuPos'
is the positive learning constant, i.e. how much is added to the weight if both
the sending and the receiving nodes are active. 'MuNeg' is the negative
learning constant, the constant associated with the LTDcomponent of Hebbian
learning. If the sending node is active and the receiving node is not, then
'MuNeg' is subtracted from the weight. 'P Creation' is the likelihood that two
nodes that are not connected get connected if both are active in a given
iteration. Connections are assumed to not exist if the weight on the connection
is equal to 0. 'P Creation' is thus the likelihood that a weight of 0 will rise
to a positive value if Hebbian learning requires that the connection learn.
'WeightMax' is the maximum value that a weight in the tract can attain. If
'WeightMax' is for example equal to 1, weights will not become higher than 1
(the minimum value a weight can have is always equal to 0). 'Dampening' is a
factor with which all weights in the tract are multiplied at the moment that
inputs are calculated. Setting 'Dampening' at a higher or lower level than 1
allows for weights to be modulated temporarily up or downwards.
On the layer level
Layers
instantiate many parameters. 'Width' and 'Height' are the layer widths and
heights. 'Active Nodes' is the nomber of nodes in the layer that is active (actually,
this is a moving average over the last iterations  with always 1/2^{nd}
determined by the last iteration, 1/4^{th} by the iteration before that
etc.). These three parameters cannot be changed. Other parameters can be
changed by the user.
'Inhibition T' and 'Inhibition tau' implement
fast and slow inhibition respectively (note that these are separate for every
layer). Their magnitude depends on the size of the layer and on the magnitudes
of the excitatory weights. Often magnitudes are around 0.05 for 'T' and 0.9 for
'tau'. 'K' is the number of nodes in the layer that should be active.
Inhibition is adjusted at every iteration to bring 'Active Nodes' closer to 'K'. 'Temperature' determines the degree
of randomness in the layer.
On the workspace level
On the
workspace level (in the browser: parameters under 'general' when one clicks
with the right muse button), there is only one parameter: 'Cycle Steps'. 'Cycle
Steps' is the number of iterations that
the model runs if you use the 'Act Cycle' method. If you intend to run many
iterations, it might be more efficient to adjust this parameter than to push or
call the 'Act Cycle' method n times.
Methods of the TraceLink
paradigm in Nutshell
♦ 'Insert
Layer' inserts a layer. As arguments you give the width and height that the
layer should have.
♦ 'Resize Layer' enables you to change the
dimensions of an existing, selected layer.
♦ 'Reset Layer Nodes' resets the activity of all nodes in the selected layer to 0,
except deactivated or clamped nodes: these are left alone by "reset".
♦ 'Insert
Tract' inserts a tract between selected layers (or, if one layer is active,
from the layer to itself).
♦ 'Delete Tract' deletes a selected tract.
♦ 'Reset Tract Conn' resets the weights of all
connections in a selected tract to 0.
♦ 'Initialize Tract' initializes the weights in the selected tract. The three arguments are
proportion, mean, and spread. 'Proportion' is the proportion of weights that
are given a value. 'Mean' and 'Spread' determine the distribution of these
weights: this distribution is a uniform distribution ranging from (MeanSpread)
to (Mean + Spread). Weights are clipped to between 0 and 'MaxWeight'.
♦ 'Act Cycle' updates the activity of nodes in
the whole network for a 'Cycle Steps' number of iterations.
♦ 'Act Step' updates the activity of nodes in the whole network for 1 iteration
♦ 'Learn' updates the weights in the whole
network on the basis of the current activation.
♦ 'Random Activity' first resets the activity
in the selected layer, and then activates k
layer nodes.
♦ 'Perturb
Tract' perturbs the weights in the selected tract. It does this by multiplying
each weight with a random factor taken from a distribution determined by a line
on the interval [0,1). You can control this distribution by giving the
arguments "slope" and "intercept" of the line (a and b in a normal line
equation). Values of greater than 1 are interpreted as 1, values lower than 0
as 0.
♦ 'Count Activity' sets the parameter 'Active
Nodes' of the selected layer to the real number of active nodes (normally it is
a moving average).
Part 2: Details of
TraceLink nodes
The model is based on binary, stochastic nodes that
fire synchronously. The firing thresholds of the nodes in a module are
controlled by a 'threshold control' mechanism: inhibition in a module is diminished
if there are not enough activated nodes (i.e., less than some target number k)
and increased if there are too many.
At each iteration, after all node activations have been updated, a learning
rule is applied to all connections. The details of these mechanisms are as
follows:
Activation rule
A node i
has an activation a_{i} that
can take on either of two values: 0 or 1. The probability that node i will 'fire'
(i.e., that its activation
becomes 1) increases with its net input, as follows:
where net_{i}
is the total input activation to node i:
net_{i}= weighted input activation  inhibition
The weighted
input activation can be written as:
where w_{ij} is the connection weight from node j to node i, a_{j} is the activation value of node j, and n is the number of nodes in the model (if there is no connection between j and i, w_{ij} is zero by default). Inhibition is discussed in the next paragraph.
The temperature parameter temp in Eq.1 controls the degree of randomness of the nodes: if temp is near zero the nodes behave as simple threshold devices, if temp is very high the role of the net input is very limited and the node will take on values 0 or 1 randomly. We used a temperature of 0.2 in all simulations.
Threshold control
The total number of activated nodes in a module (called A) is constantly monitored and firing thresholds are adjusted to ensure that this number does not wander too far from the target number k, Each module can have a different k and inhibition control in a module is independent of that in other modules. We assume that one of the functions of inhibition in the brain is to keep the level of activation (i.e., the average number of activated neurons) in a certain region constant. Furthermore, we assume that the total number of activated neurons, k, is relatively low (cf. Abeles, Vaadia, & Bergman, 1990). The effect of the inhibitory system is thus to keep the number of activated nodes A as close as possible to the target number k.
The system achieves this by constantly adjusting two thresholds T and t
. Inhibition is the sum of the fast changing threshold T multiplied by the number of active nodes A, and the slow moving threshold t
:
inhibition = TA + t
We believe that T reflects the activation of the basket cells and that it is able to change relatively quickly. In addition to the action of inhibitory cells, we assume that there are also slower working processes. These may reflect synaptic changes and growth processes. 'Slow' inhibition is modelled by the threshold t
.
The control of fast inhibition, T, is straightforward: If the total activation at time t (A_{t}) is higher than k, T is increased (more inhibition), if A_{t} is lower it is decreased. In particular, if A_{t} is much larger than k, T is increased a lot; if A_{t} is only a bit larger, T is increased a little. A very simple implementation of the approach is used here:
If A is much (i.e., more than a crit proportion) larger or smaller than k:
if A_{t} > (1+crit)k
T = T + D
_{t}
if A_{t} < (1crit)k
T = T  D
_{t}
else, if A is only a little bit larger or smaller than k:
if A_{t} > k
T = T + 1/3 D
_{t}
if A_{t} < k
T = T  1/3 D
_{t}
where crit is the criterion for deciding whether A_{t} is much larger or smaller, and D
_{t} is the change made to T (crit = 0.20, and D
_{t} = 0.01 works well for the simulations reported here). One disadvantage of this method is that T may change too quickly so that the module starts to oscillate violently. To prevent this, A_{t} is dampened by making it a moving average of the current activation and the activation of previous iterations. When A*_{t} is the current level of activation, the value used to compute both the level of inhibition A_{t}T and the change in the parameter T is:
A_{t} = 0.5A_{t1} + 0.5A*_{t}
This precedes calculation of the new threshold T.
The slow inhibition process aims to keep the 'slow threshold' t
equal to TA. When the equilibrium is disturbed, for example, if the activation is diminished due to a lesion, t
slowly decreases to a new equilibrium value. The speed of this change is determined by the parameter D
t
. Because we envision the adjustment to be slow, D
t
is chosen low (0.001). The expression for calculating t
_{t}+1 at t+1 is
t
_{t+1} = (1D
t
)t
_{t} + D
t
TA
The amount of 'fast' inhibition is bounded by a minimum value T^{min} and a maximum value T^{max}. If T < T^{min} it is set to T^{min}, and if T > T^{max} it is set to T^{max}. Similarly, t
is also kept between upper and lower bounds: if t
< t
^{min}, t
is t
^{min}; if t
> t
^{max}, t
is t
^{max}. T^{min} and t
^{mi} were set to 0. T^{max }and t
^{max} were set to such high values that they were never reached in the simulations.
Learning rule
The learning rule is a simple Hebbian rule that also allows decreases in weight, as follows:
D
w_{ij} = m
^{+} a_{i}a_{j}  m
^{} a_{i} (1  a_{j})
and
w_{ij}(t+1) = w_{ij}(t) + D
w_{ij}
where D
w_{ij} represents the weight change and m
^{} and m
^{+} represent the learning rates. Both m
^{} and m
^{+} must be larger than 0. The weights are kept within the interval [0,1] by setting w_{ij}=1 if w_{ij} >1, and w_{ij} =0 if w_{ij} <0.
Learning rate can vary rapidly due to certain central states (e.g., motivational) or slowly, (e.g., due to aging). The effects of learning rate variations are studied in various simulations in this paper, but it is kept constant from one iteration to the next (i.e., it only changes when there is a change in learning phase). In a more general version of the model the learning rate would vary continuously (cf. Murre, 1992).
