Pattern Recognition in Neural Networks with Competing Dynamics: Coexistence of Fixed-Point and Cyclic Attractors Jose ´ L. Herrera-Aguilar 1,2 *, Herna ´ n Larralde 1 , Maximino Aldana 1,3 1 Instituto de Ciencias Fı ´sicas, Universdad Nacional Auto ´ noma de Me ´xico, Cuernavaca, Morelos, Me ´ xico, 2 Facultad de Ciencias, Universidad Auto ´ noma del Estado de Morelos, Cuernavaca, Morelos, Me ´ xico, 3 FAS Center for Systems Biology and The David Rockefeller Center for Latin American Studies, Harvard University, Cambridge, Massachusetts, United States of America Abstract We study the properties of the dynamical phase transition occurring in neural network models in which a competition between associative memory and sequential pattern recognition exists. This competition occurs through a weighted mixture of the symmetric and asymmetric parts of the synaptic matrix. Through a generating functional formalism, we determine the structure of the parameter space at non-zero temperature and near saturation (i.e., when the number of stored patterns scales with the size of the network), identifying the regions of high and weak pattern correlations, the spin- glass solutions, and the order-disorder transition between these regions. This analysis reveals that, when associative memory is dominant, smooth transitions appear between high correlated regions and spurious states. In contrast when sequential pattern recognition is stronger than associative memory, the transitions are always discontinuous. Additionally, when the symmetric and asymmetric parts of the synaptic matrix are defined in terms of the same set of patterns, there is a discontinuous transition between associative memory and sequential pattern recognition. In contrast, when the symmetric and asymmetric parts of the synaptic matrix are defined in terms of independent sets of patterns, the network is able to perform both associative memory and sequential pattern recognition for a wide range of parameter values. Citation: Herrera-Aguilar JL, Larralde H, Aldana M (2012) Pattern Recognition in Neural Networks with Competing Dynamics: Coexistence of Fixed-Point and Cyclic Attractors. PLoS ONE 7(8): e42348. doi:10.1371/journal.pone.0042348 Editor: Dante R. Chialvo, National Research & Technology Council, Argentina Received February 24, 2012; Accepted July 4, 2012; Published August 10, 2012 Copyright: ß 2012 Herrera-Aguilar et al. This is an open-access article distributed under the terms of the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original author and source are credited. Funding: This work was partially supported by PAPIIT-UNAM grant IN109111 and the SEP-CONACyT grant no. 129471. MA acknowledges David Rockefeller Center at Harvard University for the Madero/ Fundacio ´n Me ´xico Fellowship. JLHA also thanks CONACyT for a Ph.D. grant under contract No. 177241 and Fundacio ´ n BBVA Bancomer-UAEM for financial support. The funders had no role in study design, data collection and analysis, decision to publish, or preparation of the manuscript. No additional external funding was received for this study. Competing Interests: The authors have declared that no competing interests exist. * E-mail: jlherrera@uach.mx Introduction Neural networks were originally developed to model the behavior of the brain. However, due to the great complexity of the brain’s neural circuitry and of the synaptic interactions, it was necessary to propose simplified models, such as the McCulloch- Pitts model [1] and the Hopfield model [2] which, although simple, still capture some important characteristics of the neuronal dynamics. One important question in this field is how and when a neural network is able to memorize a given set of patterns. Two main different mechanisms to store information in a neural network have been identified, to wit, the Associative Memory (AM) on the one hand, and the Sequential Pattern Recognition (SPR) on the other hand. The neural network performs AM when its dynamical attractors are fixed points, each corresponding to one of the patterns that we want to store in the network. This type of dynamic behavior is characterized by a symmetric interaction matrix that contains the connection strength between the neurons. Some examples of this dynamics are the Hopfield model and the Little model [2],[3], [4], [5]. Contrary to the above, in SPR the network memorizes a fixed set of patterns which are retrieved in certain order in time. From a dynamical point of view, this corresponds to a cyclic attractor consisting of the sequence of patterns stored in the network in a given order. A necessary condition for SPR to occur is that the matrix of neuron-neuron interactions has to be asymmetric. A well known example of this type of dynamics is the asymmetric Hopfield model [2], [6]. Several dynamical phases have been identified in both the AM and SPR models. Broadly speaking, these phases characterize how well the network can recognize its set of patterns, and the type of memory, i.e., whether AM or SPR. Both AM and SPR have been widely studied separately. Nonetheless, there is evidence showing that in real neural networks the synaptic connections are neither fully symmetric nor fully asymmetric [7], [8]. Rather, they can be considered as a mixture of these two cases, generating an interaction network with a complex topology. Additionally, there is evidence that the brain is capable to perform both AM and SPR [8], [9]. For instance, recalling the color of a simple object would be an example of AM, whereas recalling the digits in a phone number in the proper order would constitute an example of SPR. Since these two types of pattern retreival coexist in the brain, several authors have introduced modifications to the Hopfield model in order to obtain both types of pattern retrieval within the same network [10], [11], [12]. One approach to this problem was proposed by Coolen and Sherrington in Ref. [10]: They introduced a model in which the interaction matrix has two parts, PLOS ONE | www.plosone.org 1 August 2012 | Volume 7 | Issue 8 | e42348