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Hebbian learning reconsidered: Representation of static and dynamic objects in associative neural nets
Abstract
According to Hebb's postulate for learning, information presented to a neural net during a learning session is stored in synaptic efficacies. Long-term potentiation occurs only if the postsynaptic neuron becomes active in a time window set up by the presynaptic one. We carefully interpret and mathematically implement the Hebb rule so as to handle both stationary and dynamic objects such as single patterns and cycles. Since the natural dynamics contains a rather broad distribution of delays, the key idea is to incorporate these delays in the learning session. As theory and numerical simulations show, the resulting procedure is surprisingly robust and faithful. It also turns out the pure Hebbian learning is by selection: the network produces synaptic representations that are selected according to their resonance with the input percepts.
... Overall, the availability of top-down information in occluded fields of network area 0 is comparable to the presence of concealed scene information observed in early visual areas of humans (Smith and Muckli, 2010) that cannot be explained by purely feedforward models of perception. Unlike auto-associative models of sequential pattern-completion (Herz et al., 1989), our network forms hierarchical representations comparable to Illing et al. (2021). ...
... Other neuron-level models of invariance-learning (LeCun et al., 1989;Földiák, 1991;Rolls, 2012;Halvagal and Zenke, 2022) neither account for such feedback nor experimentally observed explicit encoding of mismatch between prediction and observation (Zmarz and Keller, 2016;Leinweber et al., 2017) and used considerably more complex learning rules requiring a larger set of assumptions (Halvagal and Zenke, 2022). Conversely, auto-associative Hopfield-type models that learn dynamic pattern completion from local learning rules (Herz et al., 1989;Brea et al., 2013) do not learn hierarchical invariant representations like the proposed model does. By solving the task of invariance learning in agreement with the generativity of sensory cortical systems, the claim for predictive coding circuits as fundamental building blocks of the brain's perceptual pathways is strengthened. ...
The ventral visual processing hierarchy of the cortex needs to fulfill at least two key functions: perceived objects must be mapped to high-level representations invariantly of the precise viewing conditions, and a generative model must be learned that allows, for instance, to fill in occluded information guided by visual experience. Here, we show how a multilayered predictive coding network can learn to recognize objects from the bottom up and to generate specific representations via a top-down pathway through a single learning rule: the local minimization of prediction errors. Trained on sequences of continuously transformed objects, neurons in the highest network area become tuned to object identity invariant of precise position, comparable to inferotemporal neurons in macaques. Drawing on this, the dynamic properties of invariant object representations reproduce experimentally observed hierarchies of timescales from low to high levels of the ventral processing stream. The predicted faster decorrelation of error-neuron activity compared to representation neurons is of relevance for the experimental search for neural correlates of prediction errors. Lastly, the generative capacity of the network is confirmed by reconstructing specific object images, robust to partial occlusion of the inputs. By learning invariance from temporal continuity within a generative model, the approach generalizes the predictive coding framework to dynamic inputs in a more biologically plausible way than self-supervised networks with non-local error-backpropagation. This was achieved simply by shifting the training paradigm to dynamic inputs, with little change in architecture and learning rule from static input-reconstructing Hebbian predictive coding networks.
... However, a considerable portion of the axons (estimated range: 20-40 percent) are very fine myelinated fibers or unmyelinated ones. These axons play quite an important role since they give rise to long delays, up to 100-200 ms (Herz et al. 1989;Miller 1994;Swadlow and Waxman 2012). In addition, there may be (post) synaptic delays, so that long-term potentiation even occurs if the signal from the source neuron has arrived some time before the target neuron becomes active. ...
... Incorporating a time lag t into even a simple neural circuit with feedback yields a scalar delay differential equation u 0 (t) ¼ βu(t) + wf (u(t t)) that defines a semiflow on the infinitedimensional phase space C ¼ C([t, 0]; ℝ) of continuous functions on the initial interval [t, 0]. It has been shown that a network of neurons with delayed signal transmission is capable of storing and retrieving spatiotemporal patterns (Herz 1996;Herz et al. 1989), but the capacity of such a network in the aforementioned studies still relies heavily on the size of the networks and the variable delays associated with different synaptic connections. In particular, if f 0 > 0 (a strictly increasing signal function) and f (0) ¼ 0 (normalization), then trajectories of such a network generically converge to a set of two stable equilibria (if w > 0, positive feedback, see Krisztin et al. (1999)) or a slowly oscillating periodic solution (if w < 0, negative feedback, see Mackey and Glass (1977); Marcus and Westervelt (1989);Walther (1995);Mallet-Paret and Nussbaum (1989)). ...
... In this study, we focus on the behaviors of a simple coupled chaotic map system that involves temporal coupling changes inspired by Hebb's rule 26) with a time delay proposed by Ito and Ohira. 21) This model is proposed as one of the simplest models of dynamical network systems that exhibits an obvious hierarchical network structure with a pacemaker element. ...
... Note that the present dynamical system fully involves the original Ito-Ohira model, and corresponds when ¼ 1. 21) In this model, w ii ¼ 0 (no self-connecting) was assumed, and the condition of P N j w ij n ¼ 1 was always satisfied for i. For the dynamics of w ij n , the present model employs a simple extension of the Hebb's rule, 26) where the state of the j-th element influences the connection from the j-th to i-th elements with a time delay of one step. Furthermore, w ij nþ1 tends to be large when x i n and x j nÀ1 are within close proximity to each other. ...
... In this study, we focus on the behaviors of a simple coupled chaotic map system that involves temporal coupling changes inspired by Hebb's rule [26], with a time delay proposed by Ito and Ohira [21]. This model was proposed as one of the simplest models of dynamical network systems that exhibits an obvious hierarchical network structure with a pacemaker element. ...
... For the dynamics of ! !" , the present model employs a simple extension of Hebb's rule [26] where the state of the -th element influences the connection from the -th to -th elements with a one-step time delay. Furthermore, !!! !" ...
In this study, we performed comprehensive morphological investigations of the spontaneous formations of effective network structures among elements in coupled logistic maps, specifically with a delayed connection change. Our proposed model showed ten states with different structural and dynamic features of the network topologies. Based on the parameter values, various stable networks, such as hierarchal networks with pacemakers or multiple layers, and a loop-shaped network were found. We also found various dynamic networks with temporal changes in the connections, which involved hidden network structures. Furthermore, we found that the shapes of the formed network structures were highly correlated to the dynamic features of the constituent elements. The present results provide diverse insights into the dynamics of neural networks and various other biological and social networks.
... The introduction of distributed delays into models of neural networks dates back to Herz et al. [8]. Differential equations with infinite delay and the dynamics have been widely studied (see Hino et al. [9] for a thorough description of the available phase spaces and some general results for this kind of equations). ...
In this paper, we approximate a nonautonomous neural network with infinite delay and a Heaviside signal function by neural networks with sigmoidal signal functions. We show that the solutions of the sigmoidal models converge to those of the Heaviside inclusion as the sigmoidal parameter vanishes. In addition, we prove the existence of pullback attractors in both cases, and the convergence of the attractors of the sigmoidal models to those of the Heaviside inclusion.
An examination of how widely distributed and specialized activities of the brain are flexibly and effectively coordinated.
A fundamental shift is occurring in neuroscience and related disciplines. In the past, researchers focused on functional specialization of the brain, discovering complex processing strategies based on convergence and divergence in slowly adapting anatomical architectures. Yet for the brain to cope with ever-changing and unpredictable circumstances, it needs strategies with richer interactive short-term dynamics. Recent research has revealed ways in which the brain effectively coordinates widely distributed and specialized activities to meet the needs of the moment. This book explores these findings, examining the functions, mechanisms, and manifestations of distributed dynamical coordination in the brain and mind across different species and levels of organization. The book identifies three basic functions of dynamic coordination: contextual disambiguation, dynamic grouping, and dynamic routing. It considers the role of dynamic coordination in temporally structured activity and explores these issues at different levels, from synaptic and local circuit mechanisms to macroscopic system dynamics, emphasizing their importance for cognition, behavior, and psychopathology.
ContributorsEvan Balaban, György Buzsáki, Nicola S. Clayton, Maurizio Corbetta, Robert Desimone, Kamran Diba, Shimon Edelman, Andreas K. Engel, Yves Fregnac, Pascal Fries, Karl Friston, Ann Graybiel, Sten Grillner, Uri Grodzinski, John-Dylan Haynes, Laurent Itti, Erich D. Jarvis, Jon H. Kaas, J.A. Scott Kelso, Peter König, Nancy J. Kopell, Ilona Kovács, Andreas Kreiter, Anders Lansner, Gilles Laurent, Jörg Lücke, Mikael Lundqvist, Angus MacDonald, Kevan Martin, Mayank Mehta, Lucia Melloni, Earl K. Miller, Bita Moghaddam, Hannah Monyer, Edvard I. Moser, May-Britt Moser, Danko Nikolic, William A. Phillips, Gordon Pipa, Constantin Rothkopf, Terrence J. Sejnowski, Steven M. Silverstein, Wolf Singer, Catherine Tallon-Baudry, Roger D. Traub, Jochen Triesch, Peter Uhlhaas, Christoph von der Malsburg, Thomas Weisswange, Miles Whittington, Matthew Wilson
Дан обзор основных идей нейросетевого моделирования когнитивных функций мозга. Описан ряд моделей нейрона (пороговый нейрон Мак‑Каллока и Питтса, нейрон с сигмоидальной функцией активации, нейрон с немонотонной функцией активации, стохастический нейрон, импульсный нейрон) и ряд нейросетевых архитектур (перцептрон, сеть обратного распространения, сеть Хопфилда, машина Больцмана). Рассмотрены структурные модели, состоящие из нескольких нейронных сетей и моделирующие функции конкретных систем мозга (гиппокамп, гиппокамп – неокортекс, префронтальная кора – базальные ганглии). Обсуждаются общие проблемы моделирования когнитивных функций мозга.
The timing of self-initiated actions shows large variability even when they are executed in stable, well-learned sequences. Could this mix of reliability and stochasticity arise within the same neural circuit? We trained rats to perform a stereotyped sequence of self-initiated actions and recorded neural ensemble activity in secondary motor cortex (M2), which is known to reflect trial-by-trial action-timing fluctuations. Using hidden Markov models, we established a dictionary between activity patterns and actions. We then showed that metastable attractors, representing activity patterns with a reliable sequential structure and large transition timing variability, could be produced by reciprocally coupling a high-dimensional recurrent network and a low-dimensional feedforward one. Transitions between attractors relied on correlated variability in this mesoscale feedback loop, predicting a specific structure of low-dimensional correlations that were empirically verified in M2 recordings. Our results suggest a novel mesoscale network motif based on correlated variability supporting naturalistic animal behavior.
Two dynamical models, proposed by Hopfield and Little to account for the collective behavior of neural networks, are analyzed. The long-time behavior of these models is governed by the statistical mechanics of infinite-range Ising spin-glass Hamiltonians. Certain configurations of the spin system, chosen at random, which serve as memories, are stored in the quenched random couplings. The present analysis is restricted to the case of a finite number p of memorized spin configurations, in the thermodynamic limit. We show that the long-time behavior of the two models is identical, for all temperatures below a transition temperature Tc. The structure of the stable and metastable states is displayed. Below Tc, these systems have 2p ground states of the Mattis type: Each one of them is fully correlated with one of the stored patterns. Below T∼0.46Tc, additional dynamically stable states appear. These metastable states correspond to specific mixings of the embedded patterns. The thermodynamic and dynamic properties of the system in the cases of more general distributions of random memories are discussed.
1
Groupe de Physique des Solides, E.N.S., 24 rue Lhomond, 75231 Paris Cedex 05, France
2
Ecole Supérieure de Physique et de Chimie Industrielles de la Ville de Paris, 10 rue Vauquelin, 75005 Paris, France
3
Unité de Neurobiologie Moléculaire and Laboratoire Associé au CNRS n° 270, Interactions Moléculaires et Cellulaires, Institut Pasteur, 25 rue du Docteur Roux, 75724 Paris Cedex 15, France
Computational properties of use to biological organisms or to the construction of computers can emerge as collective properties of systems having a large number of simple equivalent components (or neurons). The physical meaning of content-addressable memory is described by an appropriate phase space flow of the state of a system. A model of such a system is given, based on aspects of neurobiology but readily adapted to integrated circuits. The collective properties of this model produce a content-addressable memory which correctly yields an entire memory from any subpart of sufficient size. The algorithm for the time evolution of the state of the system is based on asynchronous parallel processing. Additional emergent collective properties include some capacity for generalization, familiarity recognition, categorization, error correction, and time sequence retention. The collective properties are only weakly sensitive to details of the modeling or the failure of individual devices.
We show that given certain plausible assumptions the existence of persistent states in a neural network can occur only if a certain transfer matrix has degenerate maximum eigenvalues. The existence of such states of persistent order is directly analogous to the existence of long range order in an Ising spin system; while the transition to the state of persistent order is analogous to the transition to the ordered phase of the spin system. It is shown that the persistent state is also characterized by correlations between neurons throughout the brain. It is suggested that these persistent states are associated with short term memory while the eigenvectors of the transfer matrix are a representation of long term memory. A numerical example is given that illustrates certain of these features.
There is considerable evidence, including that from single-unit classical conditioning studies, that patterns that are briefly extended in time can be learned. Learning of simultaneous patterns is usually assumed to occur by strengthening of selected synaptic connections converging on single neurons. The present paper extends this principle to the learning of patterns that are slightly extended in time. Evidence is reviewed that cortico-cortical axonal connections impose a range of conduction delays sufficient to permit temporal convergence at the single-neuron level between signals originating up to 100–200 msec apart. By selection of appropriate connections, brief temporal patterns can be represented by converging connections with different conduction delays. The Hebbian postulate for synaptic modification is sufficiently precise in time to permit such selection. Finally, this hypothesis is applied to the learning of phoneme recognition in the human forebrain, and a suggestion is made concerning the biological basis of left-hemisphere dominance for phoneme recognition.
A neural network is basically made of interconnected threshold automata i. (i=1,…., N) collecting the signals Sj produced by a set of upstream units j and delivering an output signal Si to a set of downstream units k. The evolution of the dynamic variables is driven by the following equations:
The asymmetric modification of the Hopfield model of the associative memory is considered. It is shown that the asymmetry does not change the main properties of the model, but leads to the internal nonthermal noise. The modification of the Hopfield algorithm is proposed which can be used for storing the correlated patterns and its storage capacity is estimated. The hierarchical memory model is proposed and studied.