Neural Fields

Research is continually expanding the empirical and theoretical picture of embodiment and dynamics in language. To date, however, a formalized neural dynamic framework for embodied linguistic processes has yet to emerge. To advance embodied theories of language, the present work develops a formalized neural dynamic framework of spatial language that explicitly integrates linguistic processes and dynamic sensory-motor systems. We then implement and test our spatial language architecture on a robotic platform continuously linked to real-time camera input. In a suite of tasks using everyday objects we demonstrate the framework's capacity for both contextuallydependent behavioral flexibility and the seamless integration of spatial, non-spatial, and symbolic representations. To our knowledge this is the first unified, neurally-grounded architecture integrating these processes and behaviors.

Lecture delivered for the seminar 'Spiritual and Scientific approaches to Consciousness', coordinated by Thomas G. Bever (University of Arizona), May 2nd, 2025.

The broad goal is to relate consciousness, computation, and language.

•Are there properties of computation in conscious beings that are qualitatively different from non-conscious systems?
•How can we characterise these properties?

I argue that information compression and computational oscillations (coexistence between processes of different computational complexity) are unique to conscious beings that are embedded in some physical context, because they are a consequence of how signal analysis has to work given finite processing resources (at both cognitive and physiological levels). Mapping the environment, interacting with it, making predictions about how the environment and other entities/actants will react to our actions... all that requires compression and mixed computation. A computational system that is not required to interact with a changing dynamical environment  will not develop these fundamental properties. And at the same time, a physically embedded system will necessarily develop them if it is to survive.

It is well known for a long time that, on a large-scale macroscopic scale, cortical activation states can be modeled by considering the dynamics of two non-linearly coupled populations of excitatory and inhibitory neurons , spatially organized into stereotyped modules. Considering the huge number of cells found in layered cortical areas neuronal activation is usually defined over spatially continuous domains (neural fields) as continuous distribution of neurons and synapses where each point in space corresponds to a neural population . Formally, each layer dynamics is described by integro-differential ordinary equations, where the integral kernel is a smooth function that replaces the synaptic interconnection matrix and continuous position variables replace the indices of discrete models. In this way, metric relations apply and it becomes possible to model the geometrical properties, such as spatial nearness and topology, that characterize the rich anatomy structure of cortical areas (e.g., maps, columns, dendritic and axonal arborizations, cell density, etc.) [7] [4]. By introducing a diffusive coupling into the equations that govern the punctual excitation-inhibition reactions, additional local interactions take place, providing a gradual spread of the variations of activity in space and simulates the continuity constraint of the neural field. In general, reaction-diffusion systems are natural candidates to model cortical phenomenological processes associated to sensory information processing, since their diffusion components allow for a weighted averaging of the input signal over a region of a certain extent (i.e., the receptive field), thus realizing a reduction in the complexity of the signal, while the non-linear reaction components provide a flexible medium through which all sorts of excitatory and inhibitory behaviour can be modeled.

It is well known for a long time that, on a large-scale macroscopic scale, cortical activation states can be modeled by considering the dynamics of two non-linearly coupled populations of excitatory and inhibitory neurons , spatially organized into stereotyped modules. Considering the huge number of cells found in layered cortical areas neuronal activation is usually defined over spatially continuous domains (neural fields) as continuous distribution of neurons and synapses where each point in space corresponds to a neural population . Formally, each layer dynamics is described by integro-differential ordinary equations, where the integral kernel is a smooth function that replaces the synaptic interconnection matrix and continuous position variables replace the indices of discrete models. In this way, metric relations apply and it becomes possible to model the geometrical properties, such as spatial nearness and topology, that characterize the rich anatomy structure of cortical areas (e.g., maps, columns, dendritic and axonal arborizations, cell density, etc.) [7] [4]. By introducing a diffusive coupling into the equations that govern the punctual excitation-inhibition reactions, additional local interactions take place, providing a gradual spread of the variations of activity in space and simulates the continuity constraint of the neural field. In general, reaction-diffusion systems are natural candidates to model cortical phenomenological processes associated to sensory information processing, since their diffusion components allow for a weighted averaging of the input signal over a region of a certain extent (i.e., the receptive field), thus realizing a reduction in the complexity of the signal, while the non-linear reaction components provide a flexible medium through which all sorts of excitatory and inhibitory behaviour can be modeled.

How do humans coordinate their intentions, goals and motor behaviors when performing joint action tasks? Recent experimental evidence suggests that resonance processes in the observer's motor system are crucially involved in our ability to understand actions of others', to infer their goals and even to comprehend their action-related language. In this paper, we present a control architecture for human-robot collaboration that exploits this close perceptionaction linkage as a means to achieve more natural and effi cient communication grounded in sensorimotor experiences. The architecture is formalized by a coupled system of dynamic neural fi elds representing a distributed network of neural populations that encode in their activation patterns goals, actions and shared task knowledge. We validate the verbal and nonverbal communication skills of the robot in a joint assembly task in which the human-robot team has to construct toy objects from their components. The experiments focus on the robot's capacity to anticipate the user's needs and to detect and communicate unexpected events that may occur during joint task execution.

Neural fields are neural networks which map coordinates to a desired signal. When a neural field should jointly model multiple signals, and not memorize only one, it needs to be conditioned on a latent code which describes the signal at hand. Despite being an important aspect, there has been little research on conditioning strategies for neural fields. In this work, we explore the use of neural fields as decoders for 2D semantic segmentation. For this task, we compare three conditioning methods, simple concatenation of the latent code, Feature Wise Linear Modulation (FiLM), and Cross-Attention, in conjunction with latent codes which either describe the full image or only a local region of the image. Our results show a considerable difference in performance between the examined conditioning strategies. Furthermore, we show that conditioning via Cross-Attention achieves the best results and is competitive with a CNN-based decoder for semantic segmentation.

Many of our sequential activities require that behaviors must be both precisely timed and put in the proper order. This paper presents a neuro-computational model based on the theoretical framework of Dynamic Neural Fields that supports the rapid learning and flexible adaptation of coupled order-timing representations of sequential events. A key assumption is that elapsed time is encoded in the monotonic buildup of self-stabilized neural population activity representing event memory. A stable activation gradient over subpopulations carries the information of an entire sequence. With robotics applications in mind, we test the model in simulations of a learning by observation paradigm, in which the cognitive agent first memorizes the order and relative timing of observed events and, subsequently, recalls the information from memory taking potential speed constraints into account. Model robustness is tested by systematically varying sequence complexity along the temporal and the ordinal dimension. Furthermore, an adaptation rule is proposed that allows the agent to adjust in a single trial a learned timing pattern to a changing temporal context. The simulation results are discussed with respect to our goal to endow autonomous robots with the capacity to efficiently learn complex sequences with time constraints, supporting more natural human-robot interactions.

Neural fields are neural networks which map coordinates to a desired signal. When a neural field should jointly model multiple signals, and not memorize only one, it needs to be conditioned on a latent code which describes the signal at hand. Despite being an important aspect, there has been little research on conditioning strategies for neural fields. In this work, we explore the use of neural fields as decoders for 2D semantic segmentation. For this task, we compare three conditioning methods, simple concatenation of the latent code, Feature-wise Linear Modulation (FiLM), and Cross-Attention, in conjunction with latent codes which either describe the full image or only a local region of the image. Our results show a considerable difference in performance between the examined conditioning strategies. Furthermore, we show that conditioning via Cross-Attention achieves the best results and is competitive with a CNNbased decoder for semantic segmentation.

Neural field models first appeared in the 50's, but the theory really took off in the 70's with the works of Wilson and Cowan [11, 12] and Amari [2, 1]. Neural fields are continuous networks of interacting neural masses, describing the dynamics of the cortical tissue at the population level. In this report, we study homogeneous stationary solutions (i.e independent of the spatial variable) and bump stationary solutions (i.e. localized areas of high activity) in two kinds of infinite two-dimensional neural field models composed of two neuronal layers (excitatory and inhibitory neurons). We particularly focus on bump patterns, which have been observed in the prefrontal cortex and are involved in working memory tasks [9]. We first show how to derive neural field equations from the spatialization of mesoscopic cortical column models. Then, we introduce classical techniques borrowed from Coombes [3] and Folias and Bressloff [7] to express bump solutions in a closed form and make their stability analysis. Finally we instantiate these techniques to construct stable two-dimensional bump solutions.

The problem of the genesis of oscillatory phenomena in a continuous distribution of excitatory and inhibitory neurons is addressed by introducing a neural ÿeld model of the reaction-di usion type. The presence of the di usive term, combined with the non-linear point interactions, allows the system to exhibit cooperative activation properties in both space and time. A detailed analysis of the resulting oscillatory behaviour of the model evidences its capability of generating stimulus-induced spatio-temporal coherence ÿelds, similar to the ones experimentally observed in the mammalian visual cortex. The perceptual role of the related association ÿelds, as exible media to establish feature association in the visual space, is discussed.

It is well known for a long time that, on a large-scale macroscopic scale, cortical activation states can be modeled by considering the dynamics of two non-linearly coupled populations of excitatory and inhibitory neurons [11], spatially organized into stereotyped modules. Considering the huge number of cells found in layered cortical areas neuronal activation is usually defined over spatially continuous domains (neural fields) as continuous distribution of neurons and synapses where each point in space corresponds to a neural population [6]. Formally, each layer dynamics is described by integro-differential ordinary equations, where the integral kernel is a smooth function that replaces the synaptic interconnection matrix and continuous position variables replace the indices of discrete models. In this way, metric relations apply and it becomes possible to model the geometrical properties, such as spatial nearness and topology, that characterize the rich anatomy structure of cortical areas (e.g., maps, columns, dendritic and axonal arborizations, cell density, etc.) [7] [4]. By introducing a diffusive coupling into the equations that govern the punctual excitation-inhibition reactions, additional local interactions take place, providing a gradual spread of the variations of activity in space and simulates the continuity constraint of the neural field. In general, reaction-diffusion systems are natural candidates to model cortical phenomenological processes associated to sensory information processing, since their diffusion components allow for a weighted averaging of the input signal over a region of a certain extent (i.e., the receptive field), thus realizing a reduction in the complexity of the signal, while the non-linear reaction components provide a flexible medium through which all sorts of excitatory and inhibitory behaviour can be modeled.

Many of the tasks we perform during our everyday lives are achieved through sequential execution of a set of goal-directed actions. Quite often these actions are organized hierarchically, corresponding to a nested set of goals and subgoals. Several computational models address the hierarchical execution of goaldirected actions by humans. However, the neural learning mechanisms supporting the temporal clustering of goal-directed actions in a hierarchical structure remain to a large extent unexplained. In this paper we investigate in simulations, of a dynamic neural field (DNF) model, biologically-based learning and adaptation mechanisms that can provide insight into the development of hierarchically organized internal representations of naturalistic tasks. In line with recent experimental evidence from observational learning studies, the DNF model implements the idea that prediction errors play a crucial role for grouping fine-grained events into larger units. Our ultimate goal is to use the model to endow the humanoid robot ARoS with the capability to learn hierarchies in sequential tasks, and to use that knowledge to enable efficient collaborative joint tasks with human partners. For testing the ability of the system to deal with the real-time constraints of a learning-by-demonstration paradigm we use the same assembly task from our previous work on human-robot collaboration. The model provides some insights on how hierarchically structured task representations can be learned and on how prediction errors made by the robot and signaled by the demonstrator can be used to control such process.

In this paper we study neural field models with delays which define a useful framework for modeling macroscopic parts of the cortex involving several populations of neurons. Nonlinear delayed integro-differential equations describe the spatio-temporal behavior of these fields. Using methods from the theory of delay differential equations, we show the existence and uniqueness of a solution of these equations. A Lyapunov analysis gives us sufficient conditions for the solutions to be asymptotically stable. We also present a fairly detailed study of the numerical computation of these solutions. This is, to our knowledge, the first time that a serious analysis of the problem of the existence and uniqueness of a solution of these equations has been performed. Another original contribution of ours is the definition of a Lyapunov functional and the result of stability it implies. We illustrate our numerical schemes on a variety of examples that are relevant to modeling in neuroscience.

Neural field models with delays define a useful framework for modeling macroscopic parts of the cortex involving several populations of neurons. Nonlinear delayed integrodifferential equations describe the spatio-temporal behavior of these fields. Using methods from the theory of delay differential equations, we show the existence and uniqueness of a solution of these equations. A Lyapunov analysis gives us sufficient conditions for the solutions to be asymptotically stable. We also present a study of the numerical computation of these solutions in a special case. This is, to our knowledge, the first time that a serious analysis of the problem of the existence and uniqueness of a solution of these equations has been performed. Another original contribution of ours is the definition of a Lyapunov functional and the result of stability it implies. We illustrate our work on a variety of examples that are relevant to modeling in neuroscience.

We study the neural field equations introduced by Chossat and Faugeras in [11] to model the representation and the processing of image edges and textures in the hypercolumns of the cortical area V1. The key entity, the structure tensor, intrinsically lives in a non-Euclidean, in effect hyperbolic, space. Its spatio-temporal behaviour is governed by nonlinear integro-differential equations defined on the Poincaré disc model of the two-dimensional hyperbolic space. Using methods from the theory of functional analysis we show the existence and uniqueness of a solution of these equations. In the case of stationary, i.e. time independent, solutions we perform a stability analysis which yields important results on their behavior. We also present an original study, based on non-Euclidean, hyperbolic, analysis, of a spatially localised bump solution in a limiting case. We illustrate our theoretical results with numerical simulations.

How do humans coordinate their intentions, goals and motor behaviors when performing joint action tasks? Recent experimental evidence suggests that resonance processes in the observer's motor system are crucially involved in our ability to understand actions of others', to infer their goals and even to comprehend their action-related language. In this paper, we present a control architecture for human-robot collaboration that exploits this close perceptionaction linkage as a means to achieve more natural and effi cient communication grounded in sensorimotor experiences. The architecture is formalized by a coupled system of dynamic neural fi elds representing a distributed network of neural populations that encode in their activation patterns goals, actions and shared task knowledge. We validate the verbal and nonverbal communication skills of the robot in a joint assembly task in which the human-robot team has to construct toy objects from their components. The experiments focus on the robot's capacity to anticipate the user's needs and to detect and communicate unexpected events that may occur during joint task execution.

Bayesian statistics is has been very successful in describing behavioural data on decision making and cue integration under noisy circumstances. However, it is still an open question how the human brain actually incorporates this functionality. Here we compare three ways in which Bayes rule can be implemented using neural fields. The result is a truly dynamic framework that can easily be extended by non-Bayesian mechanisms such as learning and memory.

By resorting to recent results, we show that an isomorphism exist between linguistic features of the Minimalist Program and the quantum field theory formalism of condensed matter physics. Specific linguistic features which admit a representation in terms of the many-body algebraic formalism are the unconstrained nature of recursive Merge, the operation of the Labeling Algorithm, the difference between pronounced and un-pronounced copies of elements in a sentence and the build-up of the Fibonacci sequence in the syntactic derivation of sentence structures. The collective dynamical nature of the formation process of Logical Forms leading to the individuation of the manifold of concepts and the computational self-consistency of languages are also discussed.

We propose evolutionary "analysis by synthesis" as a powerful tool in computational neuroscience. We present applications of evolution strategies to the adaptation of dynamical systems for brain modeling. First, we compare evolutionary and gradient-based optimization of dynamic neural fields on an artificial benchmark problem. Then we adjust a few-neuron model developed for explaining our recent findings in a neurobiological experiment, in which we studied the processing of temporal sequences of stimuli in the cortex.

We propose evolutionary "analysis by synthesis" as a powerful tool in computational neuroscience. We present applications of evolution strategies to the adaptation of dynamical systems for brain modeling. First, we compare evolutionary and gradient-based optimization of dynamic neural fields on an artificial benchmark problem. Then we adjust a few-neuron model developed for explaining our recent findings in a neurobiological experiment, in which we studied the processing of temporal sequences of stimuli in the cortex.

Understanding the neural field activity for realistic living systems is a challenging task in contemporary neuroscience. Neural fields have been studied and developed theoretically and numerically with considerable success over the past four decades. However, to make effective use of such models, we need to identify their constituents in practical systems. This includes the determination of model parameters and in particular the reconstruction of the underlying effective connectivity in biological tissues.In this work, we provide an integral equation approach to the reconstruction of the neural connectivity in the case where the neural activity is governed by a delay neural field equation. As preparation, we study the solution of the direct problem based on the Banach fixed-point theorem. Then we reformulate the inverse problem into a family of integral equations of the first kind. This equation will be vector valued when several neural activity trajectories are taken as input for t...

Bayesian statistics is has been very successful in describing behavioural data on decision making and cue integration under noisy circumstances. However, it is still an open question how the human brain actually incorporates this functionality. Here we compare three ways in which Bayes rule can be implemented using neural fields. The result is a truly dynamic framework that can easily be extended by non-Bayesian mechanisms such as learning and memory.

This paper reasons about the need to seek for particular kinds of models of computation that imply stronger computability than the classical models. A possible such model, constituting a chaotic dynamical system, is presented. This system, which we term as the analog shift map, when viewed as a computational model has super-Turing power and is equivalent to neural networks and the class of analog machines. This map may be appropriate to describe idealized physical phenomena.

Understanding the neural field activity for realistic living systems is a challenging task in contemporary neuroscience. Neural fields have been studied and developed theoretically and numerically with considerable success over the past four decades. However, to make effective use of such models, we need to identify their constituents in practical systems. This includes the determination of model parameters and in particular the reconstruction of the underlying effective connectivity in biological tissues.In this work, we provide an integral equation approach to the reconstruction of the neural connectivity in the case where the neural activity is governed by a delay neural field equation. As preparation, we study the solution of the direct problem based on the Banach fixed-point theorem. Then we reformulate the inverse problem into a family of integral equations of the first kind. This equation will be vector valued when several neural activity trajectories are taken as input for t...

In this paper we will try to provide arguments for the thesis that language is a physical system aiming at justificative adequacy: what architectural properties license the occurrence of certain emergent phenomena. We will claim that the derivational dynamics that can be found in language (and other systems of the mind) should be analyzed from the perspective of complex non-linear systems, as an open dynamic system. We will propose an oscillatory engine for linguistic
computations, which yields cycles as a natural emergent property given mutually incompatible tendencies between output conditions: global semantic effects and local linearization requirements. This architecture, in which structure building is conditioned by irreconciliable conditions, configures a kind of dynamical system well known in physics: a dynamical frustration. We will attempt to show that interesting effects arise when we consider that there is a dynamical frustration at the core of cognitive dynamics.

Original version of my PhD dissertation (University of Reading), a much shorter version of which (~ 60% shorter) was defended on 1/2018.

This work is a study of the nature of cognitive computation, with a focus on the relation between computation, linguistic theory, and grammar. We review traditional notions of computation and analyse their applicability to natural language, distinguishing the latter from formal languages as these are usually studied in computer science.
The main theoretical result of the thesis is that imposing a single computational template for the assignment of structural descriptions to natural language sentences, while long accepted, is both empirically inadequate and theoretically more costly than a strongly cyclic approach in which computational dependencies vary, oscillating up and down the Chomsky Hierarchy of formal grammars. Specifically, we defend the idea that such a system delivers the simplest possible structural description that captures semantic dependencies between syntactic objects in local substrings. This hypothesis will be referred to as the theory of mixed computation. The analysis of theories of computation starting with the seminal work of Alan Turing (1936) will reveal that the theory of computable functions must not be identified with the theory of effective computation; this then permits us to argue for the necessity of introducing aspects of interaction in the study of physically realized computational procedures which configure dynamical systems. We show that mixed computation lends itself naturally to modelling dynamical systems.
Empirically, we present evidence for mixed computation derived from an analysis of strong locality conditions in natural language syntax, that is, locality constraints whose violation cannot be repaired or ameliorated. Specifically, we set our focus on the Coordinate Structure Constraint (Ross, 1967), which pertains to the extraction of terms from coordinated structures and how it can be shown to arise from a mixed-computation analysis. Data from English, Spanish, and Latin support the identification of two kinds of coordinated structures: one is finite state in nature, and cannot be probed into. The other is phrase structural (context-free), and its internal complexity is visible to further syntactic operations, including extraction. We argue that data from locality conditions as well as cyclicity in natural-language syntax are accounted for naturally under a dynamical model of mixed computation.

Turing machines and Gödel numbers are important pillars of the theory of computation. Thus, any computational architecture needs to show how it could relate to Turing machines and how stable implementations of Turing computation are possible. In this chapter, we implement universal Turing computation in a neural field environment. To this end, we employ the canonical symbologram representation of a Turing machine obtained from a Gödel encoding of its symbolic repertoire and generalized shifts. The resulting nonlinear dynamical automaton (NDA) is a piecewise affine-linear map acting on the unit square that is partitioned into rectangular domains. Instead of looking at point dynamics in phase space, we then consider functional dynamics of probability distributions functions (p.d.f.s) over phase space. This is generally described by a Frobenius-Perron integral transformation that can be regarded as a neural field equation over the unit square as feature space of a dynamic field theory (DFT). Solving the Frobenius-Perron equation yields that uniform p.d.f.s with rectangular support are mapped onto uniform p.d.f.s with rectangular support, again. We call the resulting representation dynamic field automaton.

Cognitive computation, such as e.g. language processing, is conventionally regarded as Turing computation, and Turing machines can be uniquely implemented as nonlinear dynamical systems using generalized shifts and subsequent Gödel encoding of the symbolic repertoire. The resulting nonlinear dynamical automata (NDA) are piecewise affine-linear maps acting on the unit square that is partitioned into rectangular domains. Iterating a single point, i.e. a microstate, by the dynamics yields a trajectory of, in principle, infinitely many points scattered through phase space. Therefore, the NDAs microstate dynamics does not necessarily terminate in contrast to its counterpart, the symbolic dynamics obtained from the rectangular partition. In order to regain the proper symbolic interpretation, one has to prepare ensembles of randomly distributed microstates with rectangular supports. Only the resulting macrostate evolution corresponds then to the original Turing machine computation. However, the introduction of random initial conditions into a deterministic dynamics is not really satisfactory. As a possible solution for this problem we suggest a change of perspective. Instead of looking at point dynamics in phase space, we consider functional dynamics of probability distributions functions (p.d.f.s) over phase space. This is generally described by a Frobenius-Perron integral transformation that can be regarded as a neural field equation over the unit square as feature space of a dynamic field theory (DFT). Solving the Frobenius-Perron equation, yields that uniform p.d.f.s with rectangular support are mapped onto uniform p.d.f.s with rectangular support, again. Thus, the symbolically meaningful NDA macrostate dynamics becomes represented by iterated function dynamics in DFT; hence we call the resulting representation dynamic field automata.