The path-integral analysis of an associative memory model storing an infinite number of finite limit cycles

It is shown that an exact solution of the transient dynamics of an associative memory model storing an infinite number of limit cycles with l finite steps by means of the path-integral analysis. Assuming the Maxwell construction ansatz, we have succeeded in deriving the stationary state equations of the order parameters from the macroscopic recursive equations with respect to the finite-step sequence processing model which has retarded self-interactions. We have also derived the stationary state equations by means of the signal-to-noise analysis (SCSNA). The signal-to-noise analysis must assume that crosstalk noise of an input to spins obeys a Gaussian distribution. On the other hand, the path-integral method does not require such a Gaussian approximation of crosstalk noise. We have found that both the signal-to-noise analysis and the path-integral analysis give the completely same result with respect to the stationary state in the case where the dynamics is deterministic, when we assume the Maxwell construction ansatz. We have shown the dependence of storage capacity (alpha_c) on the number of patterns per one limit cycle (l). Storage capacity monotonously increases with the number of steps, and converges to alpha_c=0.269 at l ~= 10. The original properties of the finite-step sequence processing model appear as long as the number of steps of the limit cycle has order l=O(1).Comment: 24 pages, 3 figure

A Multi-Time Scale Learning Mechanism for Neuromimic Processing

Learning and representing and reasoning about temporal relations, particularly causal relations, is a deep problem in artificial intelligence (AI). Learning such representations in the real world is complicated by the fact that phenomena are subject to multiple time scale influences and may operate with a strange attractor dynamic. This dissertation proposes a new computational learning mechanism, the adaptrode, which, used in a neuromimic processing architecture may help to solve some of these problems. The adaptrode is shown to emulate the dynamics of real biological synapses and represents a significant departure from the classical weighted input scheme of conventional artificial neural networks. Indeed the adaptrode is shown, by analysis of the deep structure of real synapses, to have a strong structural correspondence with the latter in terms of multi-time scale biophysical processes. Simulations of an adaptrode-based neuron and a small network of neurons are shown to have the same learning capabilities as invertebrate animals in classical conditioning. Classical conditioning is considered a fundamental learning task in animals. Furthermore, it is subject to temporal ordering constraints that fulfill the criteria of causal relations in natural systems. It may offer clues to the learning of causal relations and mechanisms for causal reasoning. The adaptrode is shown to solve an advanced problem in classical conditioning that addresses the problem of real world dynamics. A network is able to learn multiple, contrary associations that separate in time domains, that is a long-term memory can co-exist with a short-term contrary memory without destroying the former. This solves the problem of how to deal with meaningful transients while maintaining long-term memories. Possible applications of adaptrode-based neural networks are explored and suggestions for future research are made

Theory and Practice of Computing with Excitable Dynamics

Reservoir computing (RC) is a promising paradigm for time series processing. In this paradigm, the desired output is computed by combining measurements of an excitable system that responds to time-dependent exogenous stimuli. The excitable system is called a reservoir and measurements of its state are combined using a readout layer to produce a target output. The power of RC is attributed to an emergent short-term memory in dynamical systems and has been analyzed mathematically for both linear and nonlinear dynamical systems. The theory of RC treats only the macroscopic properties of the reservoir, without reference to the underlying medium it is made of. As a result, RC is particularly attractive for building computational devices using emerging technologies whose structure is not exactly controllable, such as self-assembled nanoscale circuits. RC has lacked a formal framework for performance analysis and prediction that goes beyond memory properties. To provide such a framework, here a mathematical theory of memory and information processing in ordered and disordered linear dynamical systems is developed. This theory analyzes the optimal readout layer for a given task. The focus of the theory is a standard model of RC, the echo state network (ESN). An ESN consists of a fixed recurrent neural network that is driven by an external signal. The dynamics of the network is then combined linearly with readout weights to produce the desired output. The readout weights are calculated using linear regression. Using an analysis of regression equations, the readout weights can be calculated using only the statistical properties of the reservoir dynamics, the input signal, and the desired output. The readout layer weights can be calculated from a priori knowledge of the desired function to be computed and the weight matrix of the reservoir. This formulation explicitly depends on the input weights, the reservoir weights, and the statistics of the target function. This formulation is used to bound the expected error of the system for a given target function. The effects of input-output correlation and complex network structure in the reservoir on the computational performance of the system have been mathematically characterized. Far from the chaotic regime, ordered linear networks exhibit a homogeneous decay of memory in different dimensions, which keeps the input history coherent. As disorder is introduced in the structure of the network, memory decay becomes inhomogeneous along different dimensions causing decoherence in the input history, and degradation in task-solving performance. Close to the chaotic regime, the ordered systems show loss of temporal information in the input history, and therefore inability to solve tasks. However, by introducing disorder and therefore heterogeneous decay of memory the temporal information of input history is preserved and the task-solving performance is recovered. Thus for systems at the edge of chaos, disordered structure may enhance temporal information processing. Although the current framework only applies to linear systems, in principle it can be used to describe the properties of physical reservoir computing, e.g., photonic RC using short coherence-length light

Linear and nonlinear approaches to unravel dynamics and connectivity in neuronal cultures

[eng] In the present thesis, we propose to explore neuronal circuits at the mesoscale, an approach in which one monitors small populations of few thousand neurons and concentrates in the emergence of collective behavior. In our case, we carried out such an exploration both experimentally and numerically, and by adopting an analysis perspective centered on time series analysis and dynamical systems. Experimentally, we used neuronal cultures and prepared more than 200 of them, which were monitored using fluorescence calcium imaging. By adjusting the experimental conditions, we could set two basic arrangements of neurons, namely homogeneous and aggregated. In the experiments, we carried out two major explorations, namely development and disintegration. In the former we investigated changes in network behavior as it matured; in the latter we applied a drug that reduced neuronal interconnectivity. All the subsequent analyses and modeling along the thesis are based on these experimental data. Numerically, the thesis comprised two aspects. The first one was oriented towards a simulation of neuronal connectivity and dynamics. The second one was oriented towards the development of linear and nonlinear analysis tools to unravel dynamic and connectivity aspects of the measured experimental networks. For the first aspect, we developed a sophisticated software package to simulate single neuronal dynamics using a quadratic integrate–and–fire model with adaptation and depression. This model was plug into a synthetic graph in which the nodes of the network are neurons, and the edges connections. The graph was created using spatial embedding and realistic biology. We carried out hundreds of simulations in which we tuned the density of neurons, their spatial arrangement and the characteristics of the fluorescence signal. As a key result, we observed that homogeneous networks required a substantial number of neurons to fire and exhibit collective dynamics, and that the presence of aggregation significantly reduced the number of required neurons. For the second aspect, data analysis, we analyzed experiments and simulations to tackle three major aspects: network dynamics reconstruction using linear descriptions, dynamics reconstruction using nonlinear descriptors, and the assessment of neuronal connectivity from solely activity data. For the linear study, we analyzed all experiments using the power spectrum density (PSD), and observed that it was sufficiently good to describe the development of the network or its disintegration. PSD also allowed us to distinguish between healthy and unhealthy networks, and revealed dynamical heterogeneities across the network. For the nonlinear study, we used techniques in the context of recurrence plots. We first characterized the embedding dimension m and the time delay δ for each experiment, built the respective recurrence plots, and extracted key information of the dynamics of the system through different descriptors. Experimental results were contrasted with numerical simulations. After analyzing about 400 time series, we concluded that the degree of dynamical complexity in neuronal cultures changes both during development and disintegration. We also observed that the healthier the culture, the higher its dynamic complexity. Finally, for the reconstruction study, we first used numerical simulations to determine the best measure of ‘statistical interdependence’ among any two neurons, and took Generalized Transfer Entropy. We then analyzed the experimental data. We concluded that young cultures have a weak connectivity that increases along maturation. Aggregation increases average connectivity, and more interesting, also the assortativity, i.e. the tendency of highly connected nodes to connect with other highly connected node. In turn, this assortativity may delineates important aspects of the dynamics of the network. Overall, the results show that spatial arrangement and neuronal dynamics are able to shape a very rich repertoire of dynamical states of varying complexity.[cat] L’habilitat dels teixits neuronals de processar i transmetre informació de forma eficient depèn de les propietats dinàmiques intrínseques de les neurones i de la connectivitat entre elles. La present tesi proposa explorar diferents tècniques experimentals i de simulació per analitzar la dinàmica i connectivitat de xarxes neuronals corticals de rata embrionària. Experimentalment, la gravació de l’activitat espontània d’una població de neurones en cultiu, mitjançant una càmera ràpida i tècniques de fluorescència, possibilita el seguiment de forma controlada de l’activitat individual de cada neurona, així com la modificació de la seva connectivitat. En conjunt, aquestes eines permeten estudiar el comportament col.lectiu emergent de la població neuronal. Amb l’objectiu de simular els patrons observats en el laboratori, hem implementat un model mètric aleatori de creixement neuronal per simular la xarxa física de connexions entre neurones, i un model quadràtic d’integració i dispar amb adaptació i depressió per modelar l’ampli espectre de dinàmiques neuronals amb un cost computacional reduït. Hem caracteritzat la dinàmica global i individual de les neurones i l’hem correlacionat amb la seva estructura subjacent mitjançant tècniques lineals i no–lineals de series temporals. L’anàlisi espectral ens ha possibilitat la descripció del desenvolupament i els canvis en connectivitat en els cultius, així com la diferenciació entre cultius sans dels patològics. La reconstrucció de la dinàmica subjacent mitjançant mètodes d’incrustació i l’ús de gràfics de recurrència ens ha permès detectar diferents transicions dinàmiques amb el corresponent guany o pèrdua de la complexitat i riquesa dinàmica del cultiu durant els diferents estudis experimentals. Finalment, a fi de reconstruir la connectivitat interna hem testejat, mitjançant simulacions, diferents quantificadors per mesurar la dependència estadística entre neurona i neurona, seleccionant finalment el mètode de transferència d’entropia gereralitzada. Seguidament, hem procedit a caracteritzar les xarxes amb diferents paràmetres. Malgrat presentar certs tres de xarxes tipus ‘petit món’, els nostres cultius mostren una distribució de grau ‘exponencial’ o ‘esbiaixada’ per, respectivament, cultius joves i madurs. Addicionalment, hem observat que les xarxes homogènies presenten la propietat de disassortativitat, mentre que xarxes amb un creixent nivell d’agregació espaial presenten assortativitat. Aquesta propietat impacta fortament en la transmissió, resistència i sincronització de la xarxa

Dynamical Systems in Spiking Neuromorphic Hardware

Dynamical systems are universal computers. They can perceive stimuli, remember, learn from feedback, plan sequences of actions, and coordinate complex behavioural responses. The Neural Engineering Framework (NEF) provides a general recipe to formulate models of such systems as coupled sets of nonlinear differential equations and compile them onto recurrently connected spiking neural networks – akin to a programming language for spiking models of computation. The Nengo software ecosystem supports the NEF and compiles such models onto neuromorphic hardware. In this thesis, we analyze the theory driving the success of the NEF, and expose several core principles underpinning its correctness, scalability, completeness, robustness, and extensibility. We also derive novel theoretical extensions to the framework that enable it to far more effectively leverage a wide variety of dynamics in digital hardware, and to exploit the device-level physics in analog hardware. At the same time, we propose a novel set of spiking algorithms that recruit an optimal nonlinear encoding of time, which we call the Delay Network (DN). Backpropagation across stacked layers of DNs dramatically outperforms stacked Long Short-Term Memory (LSTM) networks—a state-of-the-art deep recurrent architecture—in accuracy and training time, on a continuous-time memory task, and a chaotic time-series prediction benchmark. The basic component of this network is shown to function on state-of-the-art spiking neuromorphic hardware including Braindrop and Loihi. This implementation approaches the energy-efficiency of the human brain in the former case, and the precision of conventional computation in the latter case

Rhythmic Action Synchronizes Memory Replay During Reinforcement Learning

Our cognitive abilities - learning from the past, sensing the current environment, planning into the future, executing an action, and infusing value into an experience - all rely on precisely timed and widespread electrical communications across neural networks. The brain’s hippocampal formation receives multimodal input, forges episodic associations, and predicts future state. Oscillating electrical bursts originating from the hippocampus, termed ‘sharp-wave ripples’ (SWR), often contain patterns of previously expressed neural spike sequences, and are necessary for certain forms of learning and memory. The discharge of SWR-replay resonates in remote parts of the brain and displays specific characteristics depending on a subject’s state of awareness and sensory context. In the sleep state, when motoric repertoire is limited, waves of breathing synchronize neural activity in several regions of the brain, including SWRs of the hippocampus. During active sensation of the awake state, cyclic licking dynamically entrains taste-reward networks in subcortical and cortical areas throughout learning. However, the neural correlates linking oromotor movements in the active learning state to the memory system of the hippocampal formation have not yet been established. Given the recurrence of SWR-replay during rhythmic ingestion of reinforcement learning and the hierarchical coupling of orofacial behaviors, we hypothesized that repeated licking could provide the oscillatory framework to synchronize memory reactivation during active learning. We approach this question with new technology development to track licking events at a reward port (P-event) during behavior on a spatial alternation task. Additionally, we developed a modular brain implant to simultaneously record from hippocampal area CA1 and medial entorhinal cortex (MEC) - interconnected brain regions that are crucial to episodic memory processing. Along with the co-modulation of individual neurons by licking and SWRs, we provide the first evidence that SWRs detected in dorsal CA1 synchronize with the phase of P-event cycle during learning. Furthermore, we confirmed that SWRs occurring during licking bouts contain neural reactivation of active navigation and trigger enhanced ripple-frequency power in downstream MEC. These results connect movement with memory and may assist in addressing abnormal ingestion behaviors that negatively affect mental or physical healt

Content-based retrieval of melodies using artificial neural networks

Human listeners are capable of spontaneously organizing and remembering a continuous stream of musical notes. A listener automatically segments a melody into phrases, from which an entire melody may be learnt and later recognized. This ability makes human listeners ideal for the task of retrieving melodies by content. This research introduces two neural networks, known as SONNETMAP and _ReTREEve, which attempt to model this behaviour. SONNET-MAP functions as a melody segmenter, whereas ReTREEve is specialized towards content-based retrieval (CBR). Typically, CBR systems represent melodies as strings of symbols drawn from a finite alphabet, thereby reducing the retrieval process to the task of approximate string matching. SONNET-MAP and ReTREEwe, which are derived from Nigrin’s SONNET architecture, offer a novel approach to these traditional systems, and indeed CBR in general. Based on melodic grouping cues, SONNETMAP segments a melody into phrases. Parallel SONNET modules form independent, sub-symbolic representations of the pitch and rhythm dimensions of each phrase. These representations are then bound using associative maps, forming a two-dimensional representation of each phrase. This organizational scheme enables SONNET-MAP to segment melodies into phrases using both the pitch and rhythm features of each melody. The boundary points formed by these melodic phrase segments are then utilized to populate the iieTREEve network. ReTREEw is organized in the same parallel fashion as SONNET-MAP. However, in addition, melodic phrases are aggregated by an additional layer; thus forming a two-dimensional, hierarchical memory structure of each entire melody. Melody retrieval is accomplished by matching input queries, whether perfect (for example, a fragment from the original melody) or imperfect (for example, a fragment derived from humming), against learned phrases and phrase sequence templates. Using a sample of fifty melodies composed by The Beatles , results show th a t the use of both pitch and rhythm during the retrieval process significantly improves retrieval results over networks that only use either pitch o r rhythm. Additionally, queries that are aligned along phrase boundaries are retrieved using significantly fewer notes than those that are not, thus indicating the importance of a human-based approach to melody segmentation. Moreover, depending on query degradation, different melodic features prove more adept at retrieval than others. The experiments presented in this thesis represent the largest empirical test of SONNET-based networks ever performed. As far as we are aware, the combined SONNET-MAP and -ReTREEue networks constitute the first self-organizing CBR system capable of automatic segmentation and retrieval of melodies using various features of pitch and rhythm

Self-organising maps : statistical analysis, treatment and applications.

This thesis presents some substantial theoretical analyses and optimal treatments of Kohonen's self-organising map (SOM) algorithm, and explores the practical application potential of the algorithm for vector quantisation, pattern classification, and image processing. It consists of two major parts. In the first part, the SOM algorithm is investigated and analysed from a statistical viewpoint. The proof of its universal convergence for any dimensionality is obtained using a novel and extended form of the Central Limit Theorem. Its feature space is shown to be an approximate multivariate Gaussian process, which will eventually converge and form a mapping, which minimises the mean-square distortion between the feature and input spaces. The diminishing effect of the initial states and implicit effects of the learning rate and neighbourhood function on its convergence and ordering are analysed and discussed. Distinct and meaningful definitions, and associated measures, of its ordering are presented in relation to map's fault-tolerance. The SOM algorithm is further enhanced by incorporating a proposed constraint, or Bayesian modification, in order to achieve optimal vector quantisation or pattern classification. The second part of this thesis addresses the task of unsupervised texture-image segmentation by means of SOM networks and model-based descriptions. A brief review of texture analysis in terms of definitions, perceptions, and approaches is given. Markov random field model-based approaches are discussed in detail. Arising from this a hierarchical self-organised segmentation structure, which consists of a local MRF parameter estimator, a SOM network, and a simple voting layer, is proposed and is shown, by theoretical analysis and practical experiment, to achieve a maximum likelihood or maximum a posteriori segmentation. A fast, simple, but efficient boundary relaxation algorithm is proposed as a post-processor to further refine the resulting segmentation. The class number validation problem in a fully unsupervised segmentation is approached by a classical, simple, and on-line minimum mean-square-error method. Experimental results indicate that this method is very efficient for texture segmentation problems. The thesis concludes with some suggestions for further work on SOM neural networks