A recurrent model of the interaction between 
Prefrontal and Inferotemporal cortex in delay 
tasks 
ALFONSO RENART, NISTOR PARGA 
Departamento de F(sica Te6rica 
Universidad Aut6noma de Madrid 
Canto Blanco, 28049 Madrid, Spain 
http://www. ft.uam.es/neurocienciaJGRUPO/grupo_english.html 
and 
EDMUND T. ROLLS 
Oxford University 
Department of Experimental Psychology 
South Parks Road, Oxford OX1 3 UD, England 
Abstract 
A very simple model of two reciprocally connected attractor neural net- 
works is studied analytically in situations similar to those encountered 
in delay match-to-sample tasks with intervening stimuli and in tasks of 
memory guided attention. The model qualitatively reproduces many of 
the experimental data on these types of tasks and provides a framework 
for the understanding of the experimental observations in the context of 
the attractor neural network scenario. 
1 Introduction 
Working memory is usually defined as the capability to actively hold information in mem- 
ory for short periods of time. In primates, visual working memory is usually studied in 
experiments in which, after the presentation of a given visual stimulus, the monkey has 
to withhold its response during a certain delay period in which no specific visual stimulus 
is shown. After the delay, another stimulus is presented and the monkey has to make a 
response which depends on the interaction between the two stimuli. In order to bridge the 
temporal gap between the stimuli, the first one has to be held in memory during the delay. 
Electrophysiological recordings in primates during the performance of this type of tasks 
has revealed that some populations of neurons in different brain areas such as prefrontal 
(PF), inferotemporal (IT) or posterior parietal (PP) cortex, maintain approximately con- 
stant firing rates during the delay periods (for a review see [ 1]) and this delay activity states 
have been postulated as the internal representations of the stimuli provoking them [2]. Al- 
though up to now most of the modeling effort regarding the operation of networks able to 
support stable delay activity states has been put in the study of uni-modular (homogeneous) 
networks, there is evidence that in order for the monkey to solve the tasks satisfactorily, the 
interaction of several different neural structures is needed. A number of studies of delay 
match-to-sample tasks with intervening stimuli in primates performed by Desimone and 
172 A. Renart, N. Parga and E. T. Rolls 
colleagues has revealed that although IT cortex supports delay activity states and shows 
memory related effects (differential responses to the same, fixed stimulus depending on its 
status on the trial, e.g. whether it matches or not the sample), it cannot, by itself, provide 
the information necessary to solve the task, as the delay activity states elicited by each of 
the stimuli in a sequence are disrupted by the input information associated with each new 
stimulus presented [3, 4, 5]. Another structure is therefore needed to store the information 
for the whole duration of the trial. PF cortex is a candidate, since it shows selective delay 
activity maintained through entire trials even with intervening stimuli [6]. A series of par- 
allel experiments by the same group on memory guided attention [7, 8] have also shown 
differential firing of IT neurons in response to the same visual stimulus shown after a delay 
(an array of figures), depending on previous information shown before the delay (one of 
the figures in the array working as a target stimulus). This evidence suggests a distributed 
memory system as the proper scenario to study working memory tasks as those described 
above. Taking into account that both IT and PF cortex are known to be able to support 
delay activity states, and that they are bi-directionally connected, in this paper we propose 
a simple model consisting of two reciprocally connected attractor neural networks to be 
identified with IT and PF cortex. Despite its simplicity, the model is able to qualitatively 
reproduce the behavior of IT and PF cortex during delay match-to-sample tasks with in- 
tervening stimuli, the behavior of IT cells during memory guided attention tasks, and to 
provide an unified picture of these experimental data in the context of associative memory 
and attractor neural networks. 
2 Model and dynamics 
The model network consists of a large number of (excitatory) neurons arranged in two 
modules. Following [9, 10], each neuron is assumed to be a dynamical element which 
transforms an incoming afferent current into an output spike rate according to a given 
transduction function. A given afferent current Iai to neuron i (i = 1,..., N) in module a 
(a - IT, PF) decays with a characteristic time constant T but increases proportionally to 
the spike rates vbj of the rest of the neurons in the network (both from inside and outside 
its module) connected to it, the contribution of each presynaptic neuron, e.g. neuron j from 
module b, being proportional to the synaptic efficacy Jf between the two. This can be 
expressed through the following equation 
/rai(t) ri(t) 
dt T 
h(eXt) 
An external current --ai from outside the network, representing the stimuli, can also 
be imposed on every neuron. Selective stimuli are modeled as proportional to the stored 
patterns, i.e. hU? 't) - h u where h, is the intensity of the external current to module a. 
'-a -- aai' 
The transduction function of the neurons transforming currents into rates has been chosen 
as a threshold hyperbolic tangent of gain G and threshold 0. 
The synaptic efficacies between the neurons of each module and between the neurons in 
different modules are respectively [ 1 l, 12] 
P 
Jo E(r/a u/-f) (r/aUJ-f) ij ; a=IT, PF 
j;,a): f(1- f)Nt =l 
(2) 
P 
 E (TIa5 -- f) (TIj -- f) V i, j ; a  b 
Jf') = f(1 - f)Nt/--1 
(3) 
Recurrent Model of lT-PF Interactions in Delay Tasks 173 
The intra-modular connections express the learning of P binary patterns {r/,U/= 0, 1, tt = 
1,..., P} by each module, each of them signaling which neurons are active in each of 
the sustained activity configurations. Each variable r/,Ui is supposed to take the values 1 
and 0 with probabilities f and (1 - f) respectively, independently across neurons and 
across patterns. The inter-modular connections reflect the temporal associations between 
the sustained activity states of each module. In this way, every stored pattern tt in the IT 
module has an associated pattern in the PF module which is labelled by the same index. 
The normalization constant Nt = N(Jo q- g) has been chosen so that the sum of the 
magnitudes of the inter- and the intra-modular connections remains constant and equal to 
1 while their relative values are varied. When this constraint is imposed the strength of 
the connections can be expressed in terms of a single independent parameter g measuring 
the relative intensity of the inter- vs. the intra-modular connections (Jo can be set equal 
to I everywhere). We will limit our study to the case where the number of stored patterns 
per module P does not increase proportionally to the size of the modules N since a large 
number of stored patterns does not seem necessary to describe the phenomenology of the 
delay match-to-sample experiments. 
Since the number of neurons in a typical network one may be interested in is very large, 
e.g.  10 5 - 10 6, the analytical treatment of the set of coupled differential equations (1) 
becomes intractable. On the other hand, when the number of neurons is large, a reliable de- 
scription of the asymptotic solutions of these equations can be found using the techniques 
of statistical mechanics [13, 9]. In this framework, instead of characterizing the states 
of the system by the state of every neuron, this characterization is performed in terms of 
macroscopic quantities called order parameters which measure and quantify some global 
properties of the network as a whole. The relevant order parameters appearing in the de- 
scription of our system are the overlaps of the state of each module with each of the stored 
patterns m, defined as: 
1 
= << f).i >>. , (4) 
i 
where the symbol <<... >>v stands for an average over the stored patterns. 
Using the free energy per neuron of the system at zero temperature r (which we do not 
write explicitly to reduce the technicalities to a minimum) we have modeled the experi- 
ments by giving the order parameters the following dynamics: 
TOrn.U = OY (5) 
Ot Ore.  
This dynamics ensures that the stationary solutions, corresponding to the values of the 
order parameters at the attractors, correspond also to minima of the free energy, and that, 
as the system evolves, the free energy is always minimized through its gradient. The time 
constant of the macroscopic dynamics is a free parameter which has been chosen equal to 
the time constant of the individual neurons, reflecting the assumption that neurons operate 
in parallel. Its value has been set to 7- = 10 ms. Equations (5) have been solved by a 
simple discretizing procedure (first order Runge-Kutta method). 
Since not all neurons in the network receive the same inputs, not all of them behave in 
the same way, i.e. have the same firing rates. In fact, the neurons in each of the module 
can be split into different sub-populations according to their state of activity in each of 
the stored patterns. The mean firing rate of the neurons in each sub-population depends 
on the particular state realized by the network (characterized by the values of the order 
parameters). Associated to each pattern there are two larger sub-populations, to be denoted 
as foreground (all active neurons) and background (all inactive neurons) of that pattern. 
174 A. Renart, N. Parga and E. T. Rolls 
The overlap with a given pattern can be expressed as the difference between the mean firing 
rate of the neurons in its foreground and its background. The average is performed over all 
other sub-populations to which each neuron in the foreground (background) may belong 
to, where the probability of a given sub-population is equal to the fraction of neurons in 
the module belonging to it (determined by the probability distribution of the stored patterns 
as given above). This partition of the neurons into sub-populations is appealing since, in 
experiments, cells are usually classified in terms of their response properties to a set of 
fixed stimuli, i.e. whether each stimulus is effective or ineffective in driving their response. 
The modeling of the different experiments proceeded according to the macroscopic dynam- 
ics (5), where each stimulus was implemented as an extra current for a desired period of 
time. 
3 Sequence with intervening stimuli 
In order to study delay match-to-sample tasks with intervening stimuli [5, 6], the module 
to be identified with IT was sequentially stimulated with external currents proportional to 
some of the stored patterns with a delay between them. To take into account the large 
fraction of PF neurons with non-selective responses to the visual stimuli (which may be 
involved in other aspects of the task different from the identification of the stimuli), and 
since the neurons in our modules are, by definition, stimulus selective (although they are 
probably connected to the non-selective neurons) a constant, non-selective current of the 
same intensity as the selective input to the IT module was applied (during the same time) 
equally to all sub-populations of the PF module. The external current to the IT module was 
stimulus selective because the fraction of IT neurons with non-selective responses to the 
visual stimuli is very small [6]. The results can be seen in Figure 1 where the sequence 
ABA with A as the sample stimulus and B as a non-matching stimulus has been studied. 
The values of the model parameters are listed in the caption. In Figure 1 a, the mean firing 
rates of the foreground populations of patterns AIT and BIT of the IT module have been 
plotted as a function of time. The main result is that, as observed in the experiments, the 
delay activity in the IT module is determined by the last stimulus presented. The delay 
activity provoked by a given stimulus is disrupted by the next, unless it corresponds to the 
same stimulus, in which case the effect of the stimulus is to increase the firing rate of the 
neurons in its foreground. We have checked that no noticeable effects occur if more non- 
matching stimuli are presented (they are all equivalent with respect to the sample) or if a 
non-match stimulus is repeated. 
If the coupling g between the modules is weak enough [12] the behavior in the PF module 
is different. This can be seen in Figure lb, where the time evolution of the mean firing rates 
of the foreground of the two associated patterns ApF and BpF stored in the PF module are 
shown. In agreement with the findings of Desimone and colleagues, the neurons in the 
PF module remain correlated with the sample for the whole trial, despite the non-selective 
signal received by all PF neurons (not only those in the foreground of the sample) and the 
fact that the selective current from the IT module tends to activate the pattern associated 
with the current stimulus. 
Desimone and colleagues [5, 6] report that the response of some neurons (not necessarily 
those with sample selective delay activity or with stimulus selective responses) in both IT 
and PF cortex to some stimuli, is larger if those stimuli are matches in their trials than if 
the same stimuli are non-matches. This has been denoted as match enhancement. In the 
present scenario the explanation is straightforward: when a stimulus is a non-match, IT and 
PF are in different states and therefore send inconsistent signals to each other. The firing 
rate of the neurons of each module is maintained in that case solely by the contribution 
to the total current coming from the recurrent collaterals. On the other hand, when the 
stimulus is the match, both modules find themselves in states associated in the synapses 
Recurrent Model of lT-PF Interactions in Delay Tasks 175 
0.8 
0.7 
0.6 
0.5 
0,4. 
0.3 
0.2 
0.1 
0 1 2 3 4 5 6 7 8 
Time (s) 
0.7 
0.6 
0.5 
0.4. 
0.3 
0.2 
0.1 
L ...............  ',... ............... j ",._. .............. 
0 1 2 3 4 5 6 7 8 9 
Time (s) 
Figure 1: (a) Mean rams in the foreground of patterns AIT (solid line) and BIT (dashed 
line) in the IT module as a function of time. (b) Same but for patterns ApF and BpF of the 
PF module. Model parameters are G = 1.3, 0 = 10 -3, f = 0.2, g = 10 -2, h = 0.13. 
Stimuli are presented during 500 ms at seconds 0, 3, and 6 following the sequence ABA. 
between the neurons connecting them, PF because it has remained that way the whole trial, 
and IT because it is driven by the current stimulus. When this happens, the contribution 
to the total current from the recurrent collaterals and from the long range afferents add up 
consistently, and the firing rate increases. In order for this explanation to hold there should 
be a correlation between the top-down input from PF and the sensory bottom-up signal to 
IT. Indeed, experimental evidence for such a correlation has very recently been found [ 14]. 
This is an important experimental finding which supports our theory. 
Looking at Figure 1, one sees that the effect is not evident in the model during the time 
of stimulus presentation, which is the period where it has been reported. The effect is, in 
fact, present, although its magnitude is too small to be noticeable in the figure. We would 
argue, however, that this quantitative difference is an artifact of the model. This is because 
the enhancement effect is very noticeable on the delay periods, where essentially the same 
neurons are active as during the stimulus presentations (i.e., where the same correlations 
between the top-down and bottom-up signals exist) but with lower firing rates. During 
stimulus presentations the firing rates are closer to the saturation regime, and therefore the 
dynamical response range of the neurons is largely reduced. 
4 Memory guided attention 
To test the differential response of cells as a function of the contents of memory, we have 
followed [7, 8] and studied a sub-population of IT cells which are simultaneously in the 
foreground of one of the patterns (AIT) and in the background of another (BIT) in the same 
conditions as the previous section (same model parameters). In Figure 2a the response 
of this sub-population as a function of time has been plotted in two different situations. 
In the first one, the effective stimulus AIT was shown first (throughout this section non 
selective stimulation of PF proceeded as in the last section) and after a delay, a stimulus 
array equal to the sum of AIT and BIT was presented. The second situation is exactly 
equal, except for the fact that the cue stimulus shown first was the ineffective stimulus BIT. 
The response of the same sub-population to the same stimulus array is totally different and 
determined by the cue stimulus: If the sub-population is in the background of the cue, its 
response is null during the trial except for the initial period of the presentation of the array. 
In accordance with the experimental observations [7, 8], its response grows initially (as 
one would expect, since during the array presentation time, stimulation is symmetric with 
176 A. Renart, N. Parga and E. T. Rolls 
respect of A and B) but is later suppressed by the top-down signal being sent by the PF 
module. This suppression provides a clear example of a situation in which the contents 
of memory (in the form of an active PF activity state) are explicitly gating the access of 
sensory information to IT, implementing a non-spatial attentional mechanism. 
0.8 
0.7 
0,6 
0.5 
0.4 
0.3 
0.2 
0.1 
0 
-0.1 
0.7 
0.6 
0.5 
0.4 
0.3 
0.2 
0.1 
1 2 3 4 5 6 0 I 2 3 
me (s) me (s) 
4 5 6 
Figure 2: (a) Mean rates as a function of time in IT neurons which are both in the fore- 
ground AIT and in the background of BIT when the cue stimulus is AIT (solid line) or BIT 
(dashed line). (b) Mean rates of the same neurons when CiT is the cue stimulus and the 
array is AIT alone (long dashed line), BiT alone (short dashed line) or the sum of AiT and 
BT (solid line). Cue present until 500 ms. Array present from 3000 ms to 3500 ms. 
Model parameters as in Figure 1 
In the model, the PF module remains in a state correlated with the cue during the whole 
trial (to our knowledge there are no measurements of PF activity during memory guided 
attention tasks) and therefore provides a persistent signal 'in the direction' of the cue which 
biases the competition between AIT and BIT established at the onset of the array. This 
is how the gating mechanism is implemented. The competitive interactions between the 
stimuli in the array are studied in Figure 2b, which is an emulation of the target-absent trials 
of [8]. In this figure, the same sub-population is studied under situations in which the cue 
stimulus is not present in the array (another one of the stored patterns, i.e. CIT) The three 
curves correspond to different arrays: The effective stimulus alone, the ineffective stimulus 
alone, and a sum of the two as in the previous experiment. In all three, the PF module 
remains in a sustained activity state correlated with CT the whole trial and therefore, since 
the patterns are independent, the signal it sends to IT is symmetric with respect of A and 
B. Thus, the response of the sub-population during the array is in this case unbiased, and 
the effect of the competitive interactions can be isolated. The result is that, as observed 
experimentally, the response to the complex array is intermediate between the one to the 
effective stimulus alone and the one to the ineffective stimulus alone. The nature of the 
competition in an attractor network like the one under study here is based on the fact that 
complex stimulus combinations are not stored in the recurrent collaterals of each module. 
These connections tend to stabilize the individual patterns which, being independent, tend 
to cancel each other when presented together. After the array is presented, the state of the 
IT module, which is correlated with CiT in the initial delay, becomes correlated with AiT 
or BiT if they are presented alone. When the array contains both of them in a symmetric 
fashion, since the sum of the patterns is not a stored pattern itself, the IT module remains 
correlated with pattern CIT due to the signal from the PF module. 
Recurrent Model of lT-PF Interactions in Delay Tasks 177 
5 Discussion 
We have proposed a toy model consisting of two reciprocally connected attractor mod- 
ules which reproduces nicely experimental observations regarding intra-trial data in delay 
match-to-sample and memory guided attention experiments in which the interaction be- 
tween IT and PF cortex is relevant. Several important issues are taken into account in the 
model: a complex interaction between the PF and IT modules resultant from the associa- 
tion of frequent patterns of activity in both modules, delay activity states in each module 
which exert mutually modulatory influences on each other, and a common substrate (we 
emphasize that the results on Sections 3 and 4 where obtained with exactly the same model 
parameters, just by changing the type of task) for the explanation of apparently diverse 
phenomena. Perception is clearly an active process which results from the complex in- 
teractions between past experience and incoming sensory information. The main goal of 
this model was to show that a very simple associational (Hebbian) pattern of connectivity 
between a perceptual module and a 'working memory' module can provide the basic in- 
gredients needed to explain coherently different experimentally found neural mechanisms 
related to this process. The model has clear limitations in terms of 'biological realism' 
which will have to be improved in order to use it to make quantitative predictions and com- 
parisons, and does not provide a complete an exhaustive account of the very complex and 
diverse phenomena in which temporo-frontal interactions are relevant (there is, for exam- 
ple, the issue of how to reset PF activity in between trials [15]). However, it is precisely the 
simplicity of the mechanism it provides and the fact that it captures the essential features of 
the experiments, despite being so simple, what makes it likely that it will remain relevant 
after being refined. 
Acknowledgements 
This work was funded by a Spanish grant PB96-0047. We acknowledge the Max Planck 
Institute for Physics of Complex Systems in Dresden, Germany, for the hospitality received 
by A.R. and N.P. during the meeting held there from March 1 to 26, 1999. 
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