A nonlinear dynamic artiﬁcial neural network model of memory

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New Ideas in Psychology ] (]]]]) ]]]–]]]

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A nonlinear dynamic artificial neural networkmodel of memory

Sylvain Chartiera,c,�, Patrice Renaudb,c, Mounir Boukadoumd

aSchool of Psychology, University of Ottawa, Montpetit 407B, 125 University Street,

Ottawa, ON, Canada K1N 6N5bUniversite du Quebec en Outaouais, CanadacInstitut Philippe Pinel de Montreal, CanadadUniversite du Quebec a Montreal, Canada

Abstract

Nonlinearity and dynamics in psychology are found in various domains such as neuroscience,

cognitive science, human development, etc. However, the models that have been proposed are mostly

of a computational nature and ignore dynamics. In those models that do include dynamic properties,

only fixed points are used to store and retrieve information, leaving many principles of nonlinear

dynamic systems (NDS) aside; for instance, chaos is often perceived as a nuisance. This paper

considers a nonlinear dynamic artificial neural network (NDANN) that implements NDS principles

while also complying with general neuroscience constraints. After a theoretical presentation,

simulation results will show that the model can exhibit multi-valued, fixed-point, region-constrained

attractors and aperiodic (including chaotic) behaviors. Because the capabilities of NDANN include

the modeling of spatiotemporal chaotic activities, it may be an efficient tool to help bridge the gap

between biological memory neural models and behavioral memory models.

Crown Copyright r 2007 Published by Elsevier Ltd. All rights reserved.

PsycINFO classification: 4160 neural networks

Keywords: Chaos theory; Cognitive science; Connectionism; Mathematical modeling; Neural networks

- see front matter Crown Copyright r 2007 Published by Elsevier Ltd. All rights reserved.

.newideapsych.2007.07.005

nding author. School of Psychology, University of Ottawa, Montpetit 407B, 125 University Street,

Canada K1N 6N5.

dress: [email protected] (S. Chartier).

this article as: Chartier, S., et al. A nonlinear dynamic artificial neural network model of memory.

in Psychology (2007), doi:10.1016/j.newideapsych.2007.07.005

www.elsevier.com/locate/newideapsych

dx.doi.org/10.1016/j.newideapsych.2007.07.005

mailto:[email protected]


ARTICLE IN PRESSS. Chartier et al. / New Ideas in Psychology ] (]]]]) ]]]–]]]2

1. Introduction

In the past, cognition has been viewed as being of computational nature. Computationalapproaches most often ignore dynamic properties. However, a multi-domain build up ofevidence exists to challenge this static view and replace it by the dynamic systemperspective (Clark, 1997; van Gelder, 1998). The application and development of thenonlinear dynamic system (NDS) has recently been receiving a lot of attention (Guastello,2000). For instance, NDS can be found in a wide range of domains that includeneuroscience (Dafilis, Liley, & Cadusch, 2001; Freeman, 1987; Korn & Faure, 2003),psychology of perception and motor coordination (DeMaris, 2000; Renaud, Bouchard, &Proulx, 2002; Renaud, Decarie, Gourd, Paquin, & Bouchard, 2003; Renaud, Singer, &Proulx, 2001; Zanone & Kelso, 1997), cognitive sciences (Erlhagen & Schoner, 2002),human development (Haken, Kelso, & Bunz, 1985; Thelen & Smith, 1994), creativeness(Guastello, 1998) and social psychology (Nowak & Vallacher, 1998). NDS is a theoreticalapproach that helps bring several spatiotemporal scales within a unified framework. Thepurpose of NDS is twofold. First, it serves as a tool to analyze data (e.g., EEG rhythms,bimanual coordination, eye movements, etc.). Second, it is used to model the differentdomains under investigation (from neuroscience to creativeness). Time and change are thetwo variables behind the strength of the NDS approach. As a result, NDS is deeplychallenging the way mental and behavioral phenomena have been studied since theinception of scientific psychology, and NDS is quickly becoming a common tool to probeand understand cognitive phenomena (e.g., memory, learning and thinking), thanks to itsability to account for their chronological dimension (Bertenthal, 2007).The way that a system changes over time is linked to the interaction between its

immediate and external surroundings. The interaction between the system and itsenvironment is essential to self-organization and complex behavior (Beer, 1995), like thedecision-making process that the system goes through when dealing with ambiguousinformation (Grossberg, 1988). If this interaction occurs under nonlinear dynamicassumptions, then a larger variety of behaviors can be exhibited (Kaplan & Glass, 1995).NDS principles are thus found in both the microworld (e.g., neural activities) and themacroworld (e.g., cognitive phenomena).In this work, a nonlinear dynamic artificial neural network (NDANN) is proposed that

exhibits NDS properties. The model is thus positioned between neural activities and low-level cognition (Fig. 1). Although it cannot all by itself connect with neuroscience andpsychology, it is a step in the direction that both worlds could be connected through theNDS perspective. Artificial neural networks (ANN) have been around since the seminalpapers of McCulloch and Pitts (1943) and Rosenblatt (1958). Although many kinds ofANN exist, cognitive psychologists usually think about the computational model that wasconceived by McClelland and Rumelhart (1986). But while ANN and NDS haveproperties in common, not all ANN models fall under the NDS perspectives (van Gelder,1998). For example, a three-layer feedforward network (e.g. Aussem, 1999; Elman et al.,1996; Mareshal & Thomas, 2006; Munakata & McClelland, 2003) has dynamic properties,but is not included in the dynamic perspective of cognition; it is associated with acomputational view of cognition instead (van Gelder, 1998).ANN considered under the umbrella of NDS can be used to achieve two goals: fit

human experimental data or implement general properties. The model presented in thispaper falls under the second goal: it implements NDS properties while being loosely

Please cite this article as: Chartier, S., et al. A nonlinear dynamic artificial neural network model of memory.

New Ideas in Psychology (2007), doi:10.1016/j.newideapsych.2007.07.005


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Nonlinear Dynamic System

Macroscopic

behaviors

(humans)

Microscopic

behaviors

(neurons)

NDANN

Model

Fig. 1. Level of modeling. The proposed model is situated between neuroscience and cognitive science modeling.

S. Chartier et al. / New Ideas in Psychology ] (]]]]) ]]]–]]] 3

constrained by neuroscience. The model is therefore part of the neurodynamic class(Haykin, 1999) where, contrary to biological neural models, units represent a smallpopulation of neurons (Skarda & Freeman, 1987). In fact, that branch of ANN is the mostpopular when considering the microscopic and macroscopic behaviors found within theNDS perspective (Haykin, 1999). One notable model is the adaptive resonance theory(ART) networks, which were proposed to solve the stability–plasticity dilemma (Carpenter& Grossberg, 1987; Grossberg, 1988). In fact, nonlinear learning principles can be tracedback to the 1960s (Grossberg, 1967). Other models, based on the distribution of thememory trace over the network, instead on a specific unit, has been around since theseventies when recurrent associative memories (RAMs) were created and then generalizedinto bidirectional associative memories (BAMs). According to Spencer and Schoner(2003), associative memories were developed around the stabilities of attractors with nounderstanding of the existing instabilities. This observation is true for earlier models (e.g.Anderson, Silverstein, Ritz, & Jones, 1977; Begin & Proulx, 1996; Hopfield, 1982; Storkey& Valabregue, 1999), but it is no longer so for newer ones (e.g. Adachi & Aihara, 1997;Aihara, Takabe, & Toyoda, 1990; Lee, 2006; Tsuda, 2001). Such models have shown thatthey can take into account classification and categorization; contrary to Prinz andBarsalou (2000). However, they are not devoid of problems. On the one hand, they are toosimple (Anderson et al., 1977; Begin & Proulx, 1996; Hopfield, 1982; Kosko, 1988); on theother, they are too complex (Adachi & Aihara, 1997; Du, Chen, Yuan, & Zhang, 2005;Imai, Osana, & Hagiwara, 2005; Lee, 2006; Lee & Farhat, 2001). A model that aims toexpress NDS properties present in both domains (depicted in Fig. 1) must be built upondynamic biological neural models (Gerstner & Kistler, 2002) while remaining as simple aspossible. The model should exhibit several properties of human cognition while stillabiding by underlying neuroscience; it should reflect in particular the rapidly changing andwidely distributed neural activation patterns that involve numerous cortical and sub-cortical regions activated in different combinations and contexts (Buchel & Friston, 2000;Sporns, Chialvo, Kaiser, & Hilgetag, 2004). Therefore, information representation withinthe network needs to be distributed and the network coding must handle both bipolar





(binary) and multi-valued stimuli (Chartier & Boukadoum, 2006a; Costantini, Casali, &Perfetti, 2003; Wang, Hwang, & Lee, 1996; Zhang & Chen, 2003; Zurada, Cloete, & vander Peol, 1996). Multi-valued coding can then be interpreted as rate coding based onpopulation averages, which is consistent with fast temporal information processing(Gerstner & Kistler, 2002) of neural activities. Consequently, coding reflects not the unit initself—as seen in localist neural networks—but rather the whole network (Adachi &Aihara, 1997; Werner, 2001).Furthermore, the static view of information representation by stable attractors (e.g.,

Hopfield-type networks) is now challenged by the dynamics of neural activities(Babloyantz & Lourenc-o, 1994; Dafilis et al., 2001; Korn & Faure, 2003) and behaviors,where spatial patterns could be stored in dynamic orbits (Tsuda, 2001). Those phenomenasuggest that, in real neural systems, information is stored and retrieved via both stable anddynamic orbit (possibly chaotic) attractors. The network proposed in this paper exhibitsboth dynamic orbit properties and fixed points. Finally, memory association and recall isnot an all-or-nothing process, but rather a progressive one (Lee, 2006). As a result, harddiscontinuities—such as the signum (sign) output function used in most Hopfield-typenetworks—as well as one-shot learning algorithms must be discarded (e.g., Grossberg,1988; Hopfield, 1982; Personnaz, Guyon, & Dreyfus, 1986; Zhao, Caceres, Damiance, &Szu, 2006).The rest of the paper is organized as follows: Section 2 presents the model and its

properties obtained from analytic and numerical results. Section 3 shows the simulationresults obtained with bipolar and multi-valued stimuli. The section also displays thesimulation results obtained under chaotic network behavior. It is followed by a discussionand conclusion.

2. Model description

2.1. Architecture

The model architecture is illustrated in Fig. 2, where x[0] and y[0] represent the initialinput-states (stimuli); t is the number of iterations over the network; and W and V areweight matrices. The network is composed of two interconnected layers that, together,allow a recurrent flow of information that is processed bidirectionally. The W layer returnsinformation to the V layer and vice versa: a function that can be viewed as a kind of top-down/bottom-up process. Like any BAM, this network can be both an associative andheteroassociative memory (Kosko, 1988). As a result, it encompasses both unsupervisedand supervised learning and is thus suitable for a general architecture under the NDSperspective. In this particular model, the two layers can be of different dimensions and,contrary to usual BAM designs, the weight matrix from one side is not necessarily thetranspose of the other side. In addition, each unit in the network corresponds to a neuralpopulation, not an individual neuron, as in a biological neural network (Skarda &Freeman, 1987), or a psychological concept.

2.2. Transmission

The output function used in our model is based on the classic Verhulst equation(Koronovskii, Trubetskov, & Khramov, 2000). This logistic growth is described by the




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Fig. 2. Network architecture.


dynamic equation

dz

dt¼ Rð1� zÞz ¼ f ðzÞ, (1)

where R is a general parameter. Eq. (1) has two fixed points: z ¼ 0 and 1. However, onlythe z ¼ 1 value is a stable fixed point. For that reason, Eq. (1) has only one attractor and itmust be modified if two attractors are desired. One way to accomplish that is to change theright-hand term to a cubic form. We then obtain

dz

dt¼ Rð1� z2Þz ¼ f ðzÞ. (2)

This last equation has three fixed points: z ¼ �1, 0 and 1, of which the two non zero onesare stable fixed points. This continuous-time differential equation can be approximated bya finite difference equation following Kaplan & Glass (1995). Let z(t) be a discrete variablefor t ¼ 0, D, 2D, ? We have

dz

dt¼ lim

D!0

zðtþ 1Þ � zðtÞ

D. (3)

If we assume D to be small but finite, the following approximation can be made:

zðtþ 1Þ � zðtÞ

D¼ f ðzðtÞÞ ) zðtþ 1Þ ¼ Df ðzðtÞÞ þ zðtÞ. (4)

With

f ðzðtÞÞ ¼ Rð1� zðtÞ2ÞzðtÞ, (5)

we obtain

zðtþ 1Þ ¼ DRð1� zðtÞ2ÞzðtÞ þ zðtÞ, (6)

where D is a small constant term. The last equation can be applied to each element of avector z. If we make the following variable changes: d ¼ DR, y(t+1) ¼ z(t+1), a(t) ¼ z(t)




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Fig. 3. Output function when d ¼ 0.2.

S. Chartier et al. / New Ideas in Psychology ] (]]]]) ]]]–]]]6

(or b(t) ¼ z(t)), and rearrange the terms of the previous equation, the following outputfunctions are obtained:

yðtþ 1Þ ¼ ðdþ 1ÞaðtÞ � daðtÞ3, (7)

xðtþ 1Þ ¼ ðdþ 1ÞbðtÞ � dbðtÞ3. (8)

In the previous equations, y(t+1) and x(t+1) represent the network outputs at time t+1;a(t) and b(t) are the corresponding usual activation functions at time t (a(t) ¼Wx(t);b(t) ¼ Vy(t)); and d is a general output parameter. As an example, Fig. 3 illustrates theshape of the output function when d ¼ 0.2 in Eq. (7). The value of d is crucial fordetermining the type of attractor in the network as the network may converge to steady,cyclic or chaotic attractors. Fig. 4 illustrates five different attractors that the outputfunction exhibits based on the d value. All of the attractors have an initial inputx(0) ¼ 0.05 and W ¼ 1; in this one-dimensional network, both x and W are scalar.Like any NDS, to guarantee that a given output converges to a fixed point, x*(t), the

slope of the derivative of the output function must be positive and less than one (Kaplan &Glass, 1995)

dyðtþ 1Þ

dWxðtÞ¼ 0oðdþ 1Þ � 3dðWxðtÞÞ2o1. (9)

This condition is satisfied when 0odo0.5 for bipolar stimuli. In that case, Wx*(t) ¼71.If Eq. (7) is expanded the relation between the input (x(t)) and output (y(t+1)) is detailed.

yðtþ 1Þ ¼ ðdþ 1ÞaðtÞ � daðtÞ3,

) yðtþ 1Þ ¼ ðdþ 1ÞðWxðtÞÞ � dðWxðtÞÞ3. ð10Þ




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Fig. 4. Attractor type in relation to the d value (a) monotonic approach to a fixed point, (b) alternate approach to

a fixed point, (c) 2-s period of oscillation, (d) positive quadrant constraint chaotic attractor, (e) chaotic attractor.





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Fig. 5. Phase portrait of the function dy/dt ¼ (1+d)Wx�d(Wx)3�x for a one-dimension network and the

corresponding vector field for a two-dimensions network.


To proceed further, Eq. (10) needs to be reformulated in a continuous time form asshown by

dy

dt¼ ðdþ 1ÞðWxÞ � dðWxÞ3 � x. (11)

For example, Fig. 5a illustrates the network’s fixed points for a one-dimensional setting.As expected, the stable fixed points are 71, while zero is unstable. Fig. 5b illustrates thevector field for a two-dimensional setting. In this case, the fixed points [71, 71]T arestable nodes, the fixed points [71, 0]T and [0, 71]T are saddle manifold points that definethe basin of attraction boundary and the fixed point [0, 0]T is an unstable node.There exists also another way to visualize the dynamics of the network using the idea of

potential energy. Energy E(y) is defined as

�dE

dy¼

dy

dt. (12)

The negative sign indicates that the state vector moves downhill in the energy landscape.This is given, using the chain rule, by the following time derivative:

dE

dt¼

dE

dy

dy

dt. (13)

Thus replacing Eq. (12) into Eq. (13) yields

dE

dt¼

dE

dy�dE

dy

� �¼ �

dE

dy

� �2

p0 (14)

Therefore E(t) decreases along trajectories or, in other words, the state vector globallyconverges towards lower energy. Equilibrium occurs at the fixed point of the vector fieldwhere local minima correspond to stable fixed points and local maxima correspond tounstable fixed points. Hence, we need to find E(y) such that

�dE=dy ¼ ðdþ 1ÞðWxÞ � dðWxÞ3 � x. (15)




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Fig. 6. Energy landscape of a one-dimensional and one-dimensional bipolar stimuli with its corresponding

contour plot.


The general solution is

EðyÞ ¼1

2yTx� yTWx� dyTWxþ

1

2dyTðWxÞ3 þ C

� �, (16)

where C is an arbitrary constant (C ¼ 0 for convenience). Similar reasoning to the oneapplied to obtain the energy function E(x) from Eq. (8) gives

EðxÞ ¼1

2xTy� xTVy� dxTVyþ

1

2dxTðVyÞ3 þ C

� �. (17)

Fig. 6 illustrates the energy function for a one-dimensional (x*(t) ¼71) and a two-dimensional network (x* ¼ [1, 1]T, [1, �1]T, [�1, 1]T, [�1, �1]T). In both cases, the numberof dimensions in both layers is equal. It is easy to see that for a one-dimensional setting, thenetwork has a double-well potential, and for a two-dimensions setting it has a quadruplepotential where the local minima correspond to the stable equilibria.

By performing a Lyapunov analysis (Kaplan & Glass, 1995), the d values that will showthe various behaviors (fixed point, cyclic and chaotic) depicted in Fig. 4 can be found. TheLyapunov exponent for the case of a one-dimensional network is approximated by

l �1

T

XT

t¼1

logdyðtþ 1Þ

dxðtÞ

��, (18)

where T is the number of network iterations, set to 10,000 to establish the approximation.In that case, the derivative term is obtained from Eq. (10), so that l is given by

l �1

T

XT

t¼1

log 1þ d� 3dxðtÞ2�� . (19)

The bifurcation diagram can also be computed. When the two diagrams are compared, it iseasy to see the link between d and the type of attractor. Fig. 7 shows that the network




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Fig. 7. Bifurcation and Lyaponov exponent diagrams as a function of d.


exhibits a monotonic approach to steady states if the value of d is between 0 and 0.5. Thebifurcation diagram shows fixed points at �1 and 1. This result is not surprising, since theweight (W) was set to 1. Finally, the proposed output function is composed of amechanism that balances the positive (d+1)ai and negative �da3i parts. Thus, a unit’soutput remains unchanged if it reaches a value of (d+1)ai�R ¼ da3i, where R is a limitwith real value (e.g. 0.7). This mechanism enables the network to exhibit multi-valuedattractor behavior (for a detailed example, see Chartier & Boukadoum, 2006a). Suchproperties contrast with the standard nonlinear output function, which can onlyexhibit bipolar attractor behavior (e.g., Anderson et al., 1977; Hopfield, 1982). It shouldbe noted that the multi-valued attractor in this model is not simply a special codingstrategy (Costantini et al., 2003; Muezzinoglu, Guzelis, & Zurada, 2003; Zhao et al., 2006)or a subdivision of the bipolar function into a staircase function (Wang et al., 1996;Zhang & Chen, 2003; Zurada et al., 1996). In those strategies, the experimenter mustknow in advance how many different real values form each stimulus to modify thearchitecture or the output function accordingly. In NDANN there is no need for theexperimenter to be involved as the network autonomously self-adapts its attractors for anygiven real values.




ARTICLE IN PRESSS. Chartier et al. / New Ideas in Psychology ] (]]]]) ]]]–]]] 11

2.3. Learning

The simplest form of weight modification is accomplished with a simple Hebbian rule,according to the equation

W ¼ YXT. (20)

In this expression, X and Y are matrices that represent the sets of bipolar pairs to beassociated, and W is the weight matrix. Eq. (20) forces the use of a one-shot learningprocess since Hebbian association is strictly additive. A more natural learning processwould make Eq. (20) incremental. But, then, the weight matrix would grow unboundedwith the repetition of the input stimuli during learning. This property may be acceptablefor orthogonal patterns, but it leads to disastrous results when the patterns are correlated.In that case, the weight matrix will be dominated by its first eigenvalue, and this will resultin recalling the same pattern whatever the input. A compromise is to use a one-shotlearning rule to limit the domination of the first eigenvalue, and to use a recurrentnonlinear output function to allow the network to filter out the different patterns duringrecall. Kosko (1988) BAM effectively used a signum output function to recall noisypatterns, despite the fact that the weight matrix developed by using Eq. (20) is not optimal.The nonlinear output function usually used by the BAM network is expressed by thefollowing equations:

yðtþ 1Þ ¼ sgn ðWxðtÞÞ (21)

and

xðtþ 1Þ ¼ sgnðWTyðtÞÞ, (22)

where sgn is the signum function defined by

sgn ðzÞ ¼

1 if z40;

0 if z ¼ 0;

�1 if zo0:

8><>: (23)

In short, by using the weight matrix defined by Eq. (20) and the output function defined byEqs. (21) and (22), the network is able to recall Y from X, and by using the weight matrixtranspose, the network is able to recall X form Y. These two processes taken togethercreate a recurrent nonlinear dynamic network with the potential to accomplish binaryassociation. However, the learning of the BAM network is performed offline and thenonlinear output function of Eqs. (21) and (22) is not used during that stage. Moreover,the network is limited to bipolar/binary patterns and, as such, cannot learn multi-valuedattractors. In addition, the network develops many spurious attractors and has limitedstorage capacity (Personnaz, Guyon, & Dreyfus, 1985). One approach to overcome theselimitations uses a projection matrix based on least mean squared error minimization(Kohonen, 1972; Personnaz et al., 1985).

W ¼ YðXTXÞ�1XT. (24)

This solution increases the storage capacity and recall performance of the network,but its learning rule, based on an inverse matrix principle, is not a local process.Several sophisticated approaches have also been proposed that modify the learning rule orcoding procedure, with the result of both increasing storage capacity and performance





(e.g., Arik, 2005; Shen & Cruz Jr., 2005). More complex learning rules such as thebackpropagation algorithm (McClelland & Rumelhart, 1986) or support vector machine(Cortes & Vapnik, 1995) could have been used. However, since the proposed model had tobe close to neuroscience, variations on Hebbian learning were favored (Gerstner & Kistler,2002). Therefore, the learning in NDANN is based on time difference Hebbian association(Chartier & Boukadoum, 2006a; Chartier & Proulx, 2005; Kosko, 1990; Sutton, 1988). It isformally expressed by the following equations:

Wðk þ 1Þ ¼WðkÞ þ Zðyð0Þ � yðtÞÞðxð0Þ þ xðtÞÞT, (25)

Vðk þ 1Þ ¼ VðkÞ þ Zðxð0Þ � xðtÞÞðyð0Þ þ yðtÞÞT, (26)

where Z represents the learning parameter. In Eqs. (25) and (26), the weight updates followthis general procedure: first, initial inputs x(0) and y(0) are fed to the network, then, thoseinputs are iterated t times through the network (Fig. 2). This results in the outputs x(t) andy(t) that are used for the weight updates. Therefore, the weights will self-stabilize when thefeedback is the same as the initial inputs (y(t) ¼ y(0) ¼ y*(t) and x(t) ¼ x(0) ¼ x*(t)); inother words, when the network has developed fixed points. The way learning works inNDANN contrasts with ART models (Grossberg, 1988) where one-shot learning occursonly when the state of the system is at a fixed point (resonance). In NDANN, the learningcauses the network to progressively develop a resonance state between the input and theoutput. Finally, since the learning explicitly incorporates the output (x(t) and y(t)), itoccurs online; thus, the learning rule is dynamically linked to the network’s output. Thiscontrasts with most BAMs, where the learning is performed solely on the activation(offline). Learning convergence is a function of the value of the learning parameter Z. Ifweight convergence is desired, Z must be set according the following condition (Chartier &Boukadoum, 2006b; Chartier & Proulx, 2005):

Zo1

2ð1� 2dÞMax½N ;M�; da1=2. (27)

3. Simulations

Several simulations were performed to illustrate the various behaviors and properties ofthe model. They were divided into two sets. The first one was devised to show thatNDANN can (1) produce the same behavior as that observed with a fixed-point associativememory, and (2) classify multi-valued stimuli which, in turn, links it to biological rate-based models. The second set of simulations dealt with dynamic orbits, where the networkstate space is different from one iteration to another. These simulations show that theproposed network can represent behavior variability without resorting to stochasticprocesses. In addition, the chaotic attractors can be bound or not depending on the desiredcontext.

3.1. Learning and recall: the fixed-point approach

The first simulation addresses the issue of iterative and convergence learning from multi-valued stimuli of different dimensions. It will show that the model can learn any kind ofstimulus, in any situation, and without need for data preprocessing (stimuli normalization




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Fig. 8. Stimuli association used for the simulation.


or orthogonalization). This feature is important since, sometimes, the entire task is donethrough preprocessing, thus making ANN use accessory.

In this simulation, the distance between the network output and the desired stimulus isobtained by a measure of error (Euclidian distance), with the number of stimuli to belearned being varied from 2 to 6. Thus, a task of learning 6 associations should be moredifficult than a task of learning 2, given the same amount of learning time (k ¼ 25 learningtrials). The stimuli are displayed in Fig. 8. The first stimulus set represents letters on a 7� 7grid, where a white pixel is assigned a value of �1 and a black pixel a value of +1. Eachletter forms a 49-dimensional bipolar vector. The second stimulus set consists of 16� 16gray-level images, with each image forming a 256-dimensional real value vector of eightlevels of gray. Therefore, the W weight matrix has 49� 256 connections and the V weightmatrix 256� 49 connections. The network task was to associate each image with itscorresponding letter (the printer image with the letter ‘‘P’’, the mailbox image with theletter ‘‘M’’, etc.). The learning parameter Z was set to 0.0025 and the output parameterd to 0.01. Both values met the requirement for weight convergence and fixed-pointdevelopment. Since the model’s learning is online and iterative, the stimuli were notpresented all at once. In order to save time, the number of iterations before each weightupdate was set to t ¼ 1. The learning followed the general procedure:

0. Initialization of weights to zero.1. Random selection of a pair following a uniform distribution.2. Stimuli iteration through the network according to Eqs. (7) and (8) (one cycle).3. Weight update according to Eqs. (25) and (26).4. Repetition of 1–3 until the desired number of learning trials is reached (k ¼ 25).

Fig. 9 illustrates the obtained results as a function of task difficulty. Easy tasks (fewerassociations) were better learned in comparison to more complex ones. In addition, aslearning increased in complexity, more and more learning interference was observed,producing greater variability in the results. To evaluate the variability for each of the




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Fig. 9. Learning curves of 2, 4 and 6 associations during 25 learning trials (a) example of a single simulation

(b) averaging over 400 simulations for the ‘‘printer association.’’


association tasks, the learning procedure was repeated 400 times. The variability wasevaluated using standard deviation. For example, in the two-association task, thevariability was given by the following average:

sdAver ¼X25

k

X2j¼1

sdjk, (28)

where k represents the learning trial, j the association pairs, and sdik is given by

sdjk ¼

P400i¼1

xijk � x�jk� �2

399(29)





with i representing the simulation trial and x the performance (1.0-error). The averagestandard deviation for the two-associations task was 0.11, while those for the four- and six-associations tasks were 0.15 and 0.20, respectively. If the number of trials is not restrainedand if there are fewer stimulus prototypes than network dimensions, then the network willbe able to learn the desired association perfectly (Chartier & Boukadoum, 2006a); afterfewer than 200 learning trials, the network could learn all of the six desired associationsperfectly (Fig. 10).

Following learning convergence, recall tests were performed to see if the networkcould show pattern completion over missing parts and noise removal, propertiesthat are generally attributed to the Gestalt theory of visual perception (Gordon, 1997).In other words, can the network develop fixed points? And is the recall processa progressive one (Lee, 2006)? Fig. 11 shows the recall output as a function of time. Usingan incomplete input (printer image), the network was able to iteratively recall theassociated letter (‘‘P’’), while also being able to reconstruct the missing parts. Contraryto signum output functions (e.g., Hopfield, 1982), it takes several iterations for a givenstimuli before converging to a fixed point. The same behavior is observed if theinitial stimulus is corrupted with noise (Fig. 12). For instance, the network effectively

Fig. 10. Weight convergence as a function of the number of learning trials.

Fig. 11. Recall output after 63% of the printer image has been removed.





completes the pattern (Fig. 13) and removes the noise (Fig. 14) if the initial condition is anoisy letter instead of the prototypes. For a comparison on the number of spuriousattractors as well as other types of BAMs on recall performance, see Chartier &Boukadoum (2006a)

Fig. 12. Recall output after 30% of the mailbox pixels have been flipped.

Fig. 13. Recall output after 43% of the letter ‘‘D’’ pixels has been removed.

Fig. 14. Recall output after 20% of the letter ‘‘L’’ pixels have been flipped.





3.2. Learning and recall: the dynamic orbit approach

This simulation was performed to see if the network could depart from the fixed-pointperspective to the dynamic orbit perspective. In order to ensure that the stimuli were wellstored, the d value was set to 0.1 during training and to a chaos leading value (1.45 or 1.65)during recall. From Fig. 7, it is easy to see that d ¼ 1.45 corresponds to a region ofrestrained chaotic behavior (Fig. 4c) and d ¼ 1.65 to an unrestrained chaotic behavior(Fig. 4d). The first simulation consisted of learning two two-dimensional bipolar stimuli:[1, 1]T and [�1, 1]T. The learning parameter (Z) was set to 0.01 and the number of learningtrials corresponded to 100. Fig. 15 shows scatter plots of the output state vectors, given thefirst stimulus (a) and the second (b) when d ¼ 1.45. As expected, the plot did not exhibitrandom behavior, but rather that of a quadratic map function. Moreover, the figure clearlyshows that the chaotic output is constrained to its corresponding quadrant (++ for thefirst stimulus and �+ for the second). Fig. 16 displays the output variations within thebasin of attraction for different random inputs. Contrary to regular BAMs, the attractorsare not the corners of a hypercube, but regions clearly delimited within quadrants. Thus,by evaluating the sign of the N quadrants it is possible to know which attractor thenetwork is on. Given the value of d, the amount of variability in the output is observed as afunction of the region volume (see the bifurcation diagram on Fig. 7); the greater thevariability, the higher the d value. Thus, even if the network behavior is chaotic, no

Fig. 15. Scatter plot of x(t+1) in function of x(t) for d ¼ 1.45.




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Fig. 16. Output variations for the quadrant restrained condition.


boundary crossing will occur while recalling an input. It is then still possible to exhibitpattern completion and reconstruction (as would be expected from a memory model).Sometimes, nonperiodic associative dynamics can be the desired behavior (Adachi &

Aihara, 1997). This behavior is characterized by the network’s ability to retrieve storedpatterns and escape from them in the transient phase. To achieve this behavior, the outputparameter is simply increased to a value equal or greater than 1.6 (Fig. 7). For instance, ifthe value of d is set to 1.65, then the output covers the entire quadrant (Figs. 17 and 18).For the sake of comparison, the next simulation used the same patterns used by Adachiand Aihara (1997) and Lee (2006). As shown in Fig. 19, the four stored stimuli are 10� 10patterns that give 100-dimensional bipolar vectors. Therefore, the W weight matrix iscomposed of 100� 100 connections and the V weight matrix of 100� 100 connections. Thelearning parameter was set to 0.0025, and the output parameter was set to 1.45 (for the firstsimulation), and to 1.65 (for the second one). Since the dimension of the network is 100and the weights are initialized at zero, the maximum squared error is thus 100. It tookNDANN less than 100 learning trials before weight convergence (Fig. 20).For the first simulation, the network chaotic output must remain bounded within a

region of the stimulus space. More precisely, each element of the output vector can varyonly within its respective quadrant. For example, after convergence, if an element isnegative, then all its successive states will be negative as well. Fig. 21 shows the networkbehavior given a noisy stimulus (30% of the pixels were flipped), when the transmissionparameter is set to d ¼ 1.45. The network progressively recalled the input into its correctquadrant. Then, the output elements always varied within their converged quadrant, likethe two-dimensional network behavior illustrated in Fig. 16, without crossing any axis. Byassigning +1 to each positive vector element and �1 to each negative element, it is easy toestablish to which particular stimulus the network converges. Thus, this behavior differsfrom the fixed-point approaches by exhibiting output variability, while sharing theirconvergence to a stable attractor from the quadrant constrained point-of-view.If the value of d is increased enough, then the network shows nonperiodic associative

dynamics. For instance, Fig. 22 displays the network output given a noise free stimulus.After a couple of iterations, the state vector leaves the attracted region and wanders fromone stored stimulus region to another. This model is therefore able to reproduce the




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Fig. 18. Output variations for the nonperiodic associative memory condition.

Fig. 17. Scatter plot of x(t+1) in function of x(t) for d ¼ 1.65.


dynamics found in Adachi and Aihara (1997). Evaluation of the stimulus distribution wasperformed by generating random patterns where each element followed a uniformdistribution xiA[�1, 1]. Each random pattern was iterated through the network for 1000cycles. The two steps were repeated with 50 different initial patterns. The proportion ofeach stimulus was then 22%, 23%, 26% and 27% (SE ¼ 0.21%). Because of this, the




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Fig. 21. Example of recall output from a noisy stimuli (30% pixel flipped).

Fig. 20. Weight convergence as a function of the number of learning trials. Since the network dimension is 100,

the maximum squared error is 100.

Fig. 19. The four stimuli used for learning.


observed distribution is statistically different from the theoretical uniform distribution. Inaddition, those memories are present in less than 15% of all the observed patterns. Thus,more than 85% of the time, the pattern is a transient memory circulating within the




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Fig. 22. Nonperiodic associative dynamics behavior (d ¼ 1.65).


stimulus-subspace. Finally, nonperiodic associative dynamics can be used as a searchprocedure for a given item. For example, if the ‘‘triangle’’ pattern symbol is the target, thenFig. 23 shows that this pattern can effectively be retrieved if the transmission parameter isset to a lower value once the network’s state is close to a match. More precisely, the figureshows that initially the transmission parameter d was set to 1.6 to allow the network to bein a nonperiodic associative state. At time t ¼ 52, the Euclidian distance from the networkstate and the stimuli was close enough (oO60) to allow lowering the transmissionparameter to d ¼ 1.45, corresponding to a quadrant constrained aperiodic state. After 10iterations (t ¼ 61), the transmission parameter d was set 0.4, corresponding to a fixed-pointbehavior. The output then rapidly converges to the desired ‘‘triangle’’ pattern.




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Fig. 23. Specific pattern retrieval in function of time. The target pattern was the ‘‘triangle’’ symbol. The minimum

distance criterion was set to O60. The transmission parameter d was lower (1.45 and 0.4) at t ¼ 52 and at t ¼ 61,

respectively.


4. Discussion and conclusion

In NDANN, the simple NDS principles (nonlinear change and time) were appliedto the output and the learning function. Those principles were implemented in a recurrentneural network and were kept as simple as possible. For the output function, a one-dimensional logistic map was employed, while for learning, it was a time differenceHebbian association. Both algorithms were linked together and subject to an onlineexchange of information that enabled the model to exhibit behaviors under theNDS perspective (e.g., learning curves, pattern completion, noise tolerance, outputdeterministic variability). The model can easily be modified to account for morebehavior. For example, if the architecture changes, then the model could be usedfor multi-step pattern learning and one-to-many associations (Chartier & Boukadoum,2006b). This temporal associative memory could be employed to model knowledgeprocessing (Osana & Hagiwara, 2000) as well as periodic behavior (Yang, Lu,Harrison, & Franc-a, 2001). Also recently, it has been shown that by only modifying thearchitecture—while keeping both learning and transmission constant—the model can





convert high-dimensional data into low-dimensional codes (Chartier, Giguere, Renaud,Proulx, & Lina, 2007). In this way, it could be potentially used for feature extraction andlearning (Hinton & Salakhutdinov, 2006).

The results with iterative online learning allow the model to find an analogyin developmental psychology, where there is progressive learning through adoption andre-adoption as a function of the interaction between stimulus relationships (Lee,2006). However, for simulation purposes, learning was made from the prototypesthemselves. In a natural setting, the categories should be extracted using a set ofexemplars instead of prototypes. In real world, each stimulus that is experiencedis different from the previous one. To overcome the possibility of an infinitenumber of stimuli, animals can regroup them in categories. Human cognition, inparticular, allows adaptation to many environments. For instance, humans learn todiscriminate almost all individual stimulations in some situations, whereas only ageneric abstraction of the category is learned in others. In some contexts, exemplarsare memorized, while in others, prototypes are abstracted and memorized. This noisylearning is possible with the incorporation of a vigilance procedure (Grossberg,1988) into distributed associative memories (Chartier, Helie, Proulx, & Boukadoum,2006; Chartier & Proulx, 1999) or by using a PCA/BAM-type architecture (Giguere,Chartier, Proulx, & Lina, 2007). Moreover, cognitive models must consider baserate learning, where the frequency of categories is employed to correctly identify anunknown item. In other contexts, however, the frequency of exemplars is used. The modelcould be easily modified to account for those different environmental biases, a propertythat winner-takes-all models, like ART, have difficulty coping with (Helie, Chartier, &Proulx, 2006).

If d is set high enough (ex. d ¼ 1.45), the network behavior will be constrainedwithin a specific region. This allowed the network to exhibit variability, while stillbeing able to show noise tolerance and pattern completion; it is an importantproperty for a dynamic associative memory. Furthermore, if d is set to a still highervalue (ex. d ¼ 1.65), then the network can display nonperiodic associative memorybehavior. The state vector in that case is never trapped in any fixed point or region;instead, it moves in non-periodic fashion from stored pattern to stored pattern.This memory searching process is clearly different from that of fixed-point associativememory models (Adachi & Aihara, 1997) and helps in the understanding of instability.The network behavior variability results from its chaotic behavior and is thusdeterministic. The network behavior was always constrained to the stimulus-subspace.Stochastic processes could also be implemented, which might then allow the network toexplore high-dimensional space through chaotic itinerancy (Kaneko & Tsuda, 2003;Tsuda, 2001) as well as chaotic wandering from a high-dimensional to a low-dimensionalattractor. This wandering could play a role in system evolvability and architecturedevelopment.

In conclusion, the present paper has shown that complex behaviors can arise fromsimple interactions between a presented network’s topology, learning and outputfunctions. The fact that the model can process multi-valued stimuli allows it to be builtin conformity with biological neural models dynamics (Gerstner & Kistler, 2002).Furthermore, the network displayed various behaviors expected within the NDS approachin psychology. As a result, the model may bring neural activities and human behaviorscloser through the satisfaction of NDS properties.





Acknowledgments

The authors are grateful to David Fung and two anonymous reviewers for their usefulhelp in reviewing this article.

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A nonlinear dynamic artiﬁcial neural network model of memory

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