On the comparison of Feller and Ornstein-Uhlenbeck models for neural activity

 Diffusion processes have been extensively used to describe membrane potential behavior. In this approach the interspike interval has a theoretical counterpart in the first-passage-time of the diffusion model employed. Since the mathematical complexity of the first-passage-time problem increases with attempts to make the models more realistic it seems useful to compare the features of different models in order to highlight their relative performance. In this paper we compare the Feller and Ornstein–Uhlenbeck models under three different criteria derived from the level of information available about their parameters. We conclude that the Feller model is preferable when complete knowledge of the characterizing parameters is assumed. On the other hand, when only limited information about the parameters is available, such as the mean firing time and the histogram shape, no advantage arises from using this more complex model. Received: 8 November 1994/Accepted in revised form : 23 May 1995     

A Granger causality measure for point process models of ensemble neural spiking activity

The ability to identify directional interactions that occur among multiple neurons in the brain is crucial to an understanding of how groups of neurons cooperate in order to generate specific brain functions. However, an optimal method of assessing these interactions has not been established. Granger causality has proven to be an effective method for the analysis of the directional interactions between multiple sets of continuous-valued data, but cannot be applied to neural spike train recordings due to their discrete nature. This paper proposes a point process framework that enables Granger causality to be applied to point process data such as neural spike trains. The proposed framework uses the point process likelihood function to relate a neuron's spiking probability to possible covariates, such as its own spiking history and the concurrent activity of simultaneously recorded neurons. Granger causality is assessed based on the relative reduction of the point process likelihood of one neuron obtained excluding one of its covariates compared to the likelihood obtained using all of its covariates. The method was tested on simulated data, and then applied to neural activity recorded from the primary motor cortex (MI) of a Felis catus subject. The interactions present in the simulated data were predicted with a high degree of accuracy, and when applied to the real neural data, the proposed method identified causal relationships between many of the recorded neurons. This paper proposes a novel method that successfully applies Granger causality to point process data, and has the potential to provide unique physiological insights when applied to neural spike trains.     

A qualitative comparison of some diffusion models for neural activity via stochastic ordering

A number of diffusion processes have been proposed as a continuous analog of Stein's model for the subthreshold membrane potential of a neuron. Interspike intervals are then described as the first-passage-time of the corresponding diffusion model through a suitable threshold. Various biological considerations suggest the use of more sophisticated models in lieu of the Ornstein-Uhlenbeck model. However, the advantages of the additional complexity are not always clear. Comparisons among different models generally use numerical methods in specific examples without a general sensitivity analysis on the role of the model parameters. Here, we compare the distribution of interspike intervals from different models using the method of stochastic ordering. The qualitative comparison of the role of each parameter extends the results obtained from numerical simulations. One result on neurons with high positive net excitation is that the reversal potential models considered do not greatly differ from the Ornstein-Uhlenbeck model. For neurons with increased inhibition, the models give greater differences among the interspike interval distributions. In particular, when the mean trajectories are matched, the Feller model gives shorter times than the Ornstein-Uhlenbeck model but longer times than our double reversal potential model. Received: 5 August 1999 / Accepted in revised form: 8 May 2000     

Oscillatory models of the hippocampus: A study of spatio-temporal patterns of neural activity

Spatial patterns of theta-rhythm activity in oscillatory models of the hippocampus are studied here using canonical models for both Hodgkin's class-1 and class-2 excitable neuronal systems. Dynamics of these models are studied in both the frequency domain, to determine phase-locking patterns, and in the time domain, to determine the amplitude responses resulting from phase-locking patterns. Computer simulations presented here demonstrate that phase deviations (timings) between inputs from the medial septum and the entorhinal cortex can create spatial patterns of theta-rhythm phase-locking. In this way, we show that the timing of inputs (not only their frequencies alone) can encode specific patterns of theta-rhythm activity. This study suggests new experiments to determine temporal and spatial synchronization. Received: 31 July 1998 /Accepted in revised form: 20 April 1999     

Physical models of neural networks

The state of art in computer modelling of neural networks with associative memory is reviewed. The available experimental data are considered on learning and memory of small neural systems, on isolated synapses and on molecular level. Computer simulations demonstrate that realistic models of neural ensembles exhibit properties which can be interpreted as image recognition, categorization, learning, prototype forming, etc. A bilayer model of associative neural network is proposed. One layer corresponds to the short-term memory, the other one to the long-term memory. Patterns are stored in terms of the synaptic strength matrix. We have studied the relaxational dynamics of neurons firing and suppression within the short-term memory layer under the influence of the long-term memory layer. The interaction among the layers has found to create a number of novel stable states which are not the learning patterns. These synthetic patterns may consist of elements belonging to different non-intersecting learning patterns. Within the framework of a hypothesis of selective and definite coding of images in brain one can interpret the observed effect as the "idea? generating" process.     

Some elementary considerations of neural models

The outputs of nervous systems (as expressed in motor activity) are viewed as mathematical transformations on the inputs which enter via the sensory nerves. Simple nerve-ganglion models are exhibited which theoretically account for the arithmetic computations necessary to expedite such transformations.     

Simple neural models of classical conditioning

A systematic study of the necessary and sufficient ingredients of a successful model of classical conditioning is presented. Models are constructed along the lines proposed by Gelperin, Hopfield, and Tank, who showed that many conditioning phenomena could be reproduced in a model using non-trivial distributed representations of the sensory stimuli. The additional phenomena of extinction and blocking are found to be obtainable by generalizing the Hebbian learning algorithm, rather than by additional complications in the hardware. The most successful algorithms have a minimal number of adjustable parameters, and require only local-time information about the level of postsynaptic activity. The proper behavior of these algorithms is verified by both simple analytic arguments and by direct numerical simulation. Certain detailed assumptions concerning the distributed sensory representations are also found to have a surprising degree of importance.     

Cyclic modes in neural net models

Neural network models with possible cross-coupling from every neuron to every other neuron are condisered. The all-or-none law is assumed for firing of neurons. A network is shown to have a set of latent cyclic modes. If the net is stimulated briefly, it will subsequently either return to quiescence or settle into periodic activity in one of its cyclic modes. Realization of cyclic modes and analysis of nets are discussed. A learning rule for adjustment of synaptic strengths is presented.     

Parameter estimation for bursting neural models

This paper presents work on parameter estimation methods for bursting neural models. In our approach we use both geometrical features specific to bursting, as well as general features such as periodic orbits and their bifurcations. We use the geometry underlying bursting to introduce defining equations for burst initiation and termination, and restrict the estimation algorithms to the space of bursting periodic orbits when trying to fit periodic burst data. These geometrical ideas are combined with automatic differentiation to accurately compute parameter sensitivities for the burst timing and period. In addition to being of inherent interest, these sensitivities are used in standard gradient-based optimization algorithms to fit model burst duration and period to data. As an application, we fit Butera et al.'s (Journal of Neurophysiology 81, 382-397, 1999) model of preB?tzinger complex neurons to empirical data both in control conditions and when the neuromodulator norepinephrine is added (Viemari and Ramirez, Journal of Neurophysiology 95, 2070-2082, 2006). The results suggest possible modulatory mechanisms in the preB?tzinger complex, including modulation of the persistent sodium current.     

Natural patterns of neural activity

Physiological mechanisms of neuronal information processing have been shaped during evolution by a continual interplay between organisms and their sensory surroundings. Thus, when asking for the functional significance of such mechanisms, the natural conditions under which they operate must be considered. This has been done successfully in several studies that employ sensory stimulation under in vivo conditions. These studies address the question of how physiological mechanisms within neurons are properly adjusted to the characteristics of natural stimuli and to the demands imposed on the system being studied. Results from diverse animal models show how neurons exploit natural stimulus statistics efficiently by utilizing specific filtering capacities. Mechanisms that allow neurons to adapt to the currently relevant range from an often immense stimulus spectrum are outlined, and examples are provided that suggest that information transfer between neurons is shaped by the system-specific computational tasks in the behavioral context.     

Using theoretical models to analyse neural development

The development of the nervous system is an extremely complex and dynamic process. Through the continuous interplay of genetic information and changing intra- and extracellular environments, the nervous system constructs itself from precursor cells that divide and form neurons, which migrate, differentiate and establish synaptic connections. Our understanding of neural development can be greatly assisted by mathematical and computational modelling, because it allows us to bridge the gap between system-level dynamics and the lower level cellular and molecular processes. This Review shows the potential of theoretical models to examine many aspects of neural development.     

Improved neuronal models for studying neural networks

Previous neuronal models used for the study of neural networks are considered. Equations are developed for a model which includes: 1) a normalized range of firing rates with decreased sensitivity at large excitatory or large inhibitory input levels, 2) a single rate constant for the increase in firing rate following step changes in the input, 3) one or more rate constants, as required to fit experimental data for the adaptation of firing rates to maintained inputs. Computed responses compare well with the types of neuronal responses observed experimentally. Depending on the parameters, overdamped increases and decreases, damped oscillatory or maintained oscillatory changes in firing rate are observed to step changes in the input. The integrodifferential equations describing the neuronal models can be represented by a set of first-order differential equations. Steady-state solutions for these equations can be obtained for constant inputs, as well as the stability of the solutions to small perturbations. The linear frequency response function is derived for sufficiently small time-varying inputs. The linear responses are also compared with the computed solutions for larger non-linear responses.     

Evolutionary optimization and neural network models of behavior

One of the main challenges to the adaptionist program in general and the use of optimization models in behavioral and evolutionary ecology, in particular, is that organisms are so constrained' by ontogeny and phylogeny that they may not be able to attain optimal solutions, however those are defined. This paper responds to the challenge through the comparison of optimality and neural network models for the behavior of an individual polychaete worm. The evolutionary optimization model is used to compute behaviors (movement in and out of a tube) that maximize a measure of Darwinian fitness based on individual survival and reproduction. The neural network involves motor, sensory, energetic reserve and clock neuronal groups. Ontogeny of the neural network is the change of connections of a single individual in response to its experiences in the environment. Evolution of the neural network is the natural selection of initial values of connections between groups and learning rules for changing connections. Taken together, these can be viewed as design parameters. The best neural networks have fitnesses between 85% and 99% of the fitness of the evolutionary optimization model. More complicated models for polychaete worms are discussed. Formulation of a neural network model for host acceptance decisions by tephritid fruit flies leads to predictions about the neurobiology of the flies. The general conclusion is that neural networks appear to be sufficiently rich and plastic that even weak evolution of design parameters may be sufficient for organisms to achieve behaviors that give fitnesses close to the evolutionary optimal fitness, particularly if the behaviors are relatively simple.     

From cognitive to neural models of working memory

Working memory refers to the temporary retention of information that was just experienced or just retrieved from long-term memory but no longer exists in the external environment. These internal representations are short-lived, but can be stored for longer periods of time through active maintenance or rehearsal strategies, and can be subjected to various operations that manipulate the information in such a way that makes it useful for goal-directed behaviour. Empirical studies of working memory using neuroscientific techniques, such as neuronal recordings in monkeys or functional neuroimaging in humans, have advanced our knowledge of the underlying neural mechanisms of working memory. This rich dataset can be reconciled with behavioural findings derived from investigating the cognitive mechanisms underlying working memory. In this paper, I review the progress that has been made towards this effort by illustrating how investigations of the neural mechanisms underlying working memory can be influenced by cognitive models and, in turn, how cognitive models can be shaped and modified by neuroscientific data. One conclusion that arises from this research is that working memory can be viewed as neither a unitary nor a dedicated system. A network of brain regions, including the prefrontal cortex (PFC), is critical for the active maintenance of internal representations that are necessary for goal-directed behaviour. Thus, working memory is not localized to a single brain region but probably is an emergent property of the functional interactions between the PFC and the rest of the brain.