Modeling brain resonance phenomena using a neural mass model

Stimulation with rhythmic light flicker (photic driving) plays an important role in the diagnosis of schizophrenia, mood disorder, migraine, and epilepsy. In particular, the adjustment of spontaneous brain rhythms to the stimulus frequency (entrainment) is used to assess the functional flexibility of the brain. We aim to gain deeper understanding of the mechanisms underlying this technique and to predict the effects of stimulus frequency and intensity. For this purpose, a modified Jansen and Rit neural mass model (NMM) of a cortical circuit is used. This mean field model has been designed to strike a balance between mathematical simplicity and biological plausibility. We reproduced the entrainment phenomenon observed in EEG during a photic driving experiment. More generally, we demonstrate that such a single area model can already yield very complex dynamics, including chaos, for biologically plausible parameter ranges. We chart the entire parameter space by means of characteristic Lyapunov spectra and Kaplan-Yorke dimension as well as time series and power spectra. Rhythmic and chaotic brain states were found virtually next to each other, such that small parameter changes can give rise to switching from one to another. Strikingly, this characteristic pattern of unpredictability generated by the model was matched to the experimental data with reasonable accuracy. These findings confirm that the NMM is a useful model of brain dynamics during photic driving. In this context, it can be used to study the mechanisms of, for example, perception and epileptic seizure generation. In particular, it enabled us to make predictions regarding the stimulus amplitude in further experiments for improving the entrainment effect.     

Modeling the effect of sleep regulation on a neural mass model

In mammals, sleep is categorized by two main sleep stages, rapid eye movement (REM) and non-REM (NREM) sleep that are known to fulfill different functional roles, the most notable being the consolidation of memory. While REM sleep is characterized by brain activity similar to wakefulness, the EEG activity changes drastically with the emergence of K-complexes, sleep spindles and slow oscillations during NREM sleep. These changes are regulated by circadian and ultradian rhythms, which emerge from an intricate interplay between multiple neuronal populations in the brainstem, forebrain and hypothalamus and the resulting varying levels of neuromodulators. Recently, there has been progress in the understanding of those rhythms both from a physiological as well as theoretical perspective. However, how these neuromodulators affect the generation of the different EEG patterns and their temporal dynamics is poorly understood. Here, we build upon previous work on a neural mass model of the sleeping cortex and investigate the effect of those neuromodulators on the dynamics of the cortex and the corresponding transition between wakefulness and the different sleep stages. We show that our simplified model is sufficient to generate the essential features of human EEG over a full day. This approach builds a bridge between sleep regulatory networks and EEG generating neural mass models and provides a valuable tool for model validation.     

Slow oscillations in blood pressure via a nonlinear feedback model

Blood pressure is well established to contain a potential oscillation between 0.1 and 0.4 Hz, which is proposed to reflect resonant feedback in the baroreflex loop. A linear feedback model, comprising delay and lag terms for the vasculature, and a linear proportional derivative controller have been proposed to account for the 0.4-Hz oscillation in blood pressure in rats. However, although this model can produce oscillations at the required frequency, some strict relationships between the controller and vasculature parameters must be true for the oscillations to be stable. We developed a nonlinear model, containing an amplitude-limiting nonlinearity that allows for similar oscillations under a very mild set of assumptions. Models constructed from arterial pressure and sympathetic nerve activity recordings obtained from conscious rabbits under resting conditions suggest that the nonlinearity in the feedback loop is not contained within the vasculature, but rather is confined to the central nervous system. The advantage of the model is that it provides for sustained stable oscillations under a wide variety of situations even where gain at various points along the feedback loop may be altered, a situation that is not possible with a linear feedback model. Our model shows how variations in some of the nonlinearity characteristics can account for growth or decay in the oscillations and situations where the oscillations can disappear altogether. Such variations are shown to accord well with observed experimental data. Additionally, using a nonlinear feedback model, it is straightforward to show that the variation in frequency of the oscillations in blood pressure in rats (0.4 Hz), rabbits (0.3 Hz), and humans (0.1 Hz) is primarily due to scaling effects of conduction times between species.     

Modeling neural mechanisms of vertebrate habituation: Locus specificity and pattern discrimination

A critical problem in neurobiology is to explain how the central nervous system coordinates pattern discrimination and locus specificity in learning. This problem is investigated in anuran amphibians who demonstrate both locus specificity and pattern discrimination in visual habituation. A neural mechanism is proposed whereby neural circuitry for pattern discrimination is shared by a spatial memory system. Such learning processes are argued to occur in the medial pallium (MP), the anuran's homolog of mammalian hippocampus. Necessary mapping from the shared network to spatial memory is set up by a mechanism that forms topographical connections, with desired orientation determined by activity gradient in presynaptic and postsynaptic layers. The model of MP is tested on both locus and stimulus specific habituation, which involve short-term as well as long-term synaptic plasticity. Successful modeling yields a set of predictions concerning MP organization and learning properties.     

Winner-take-all selection in a neural system with delayed feedback

We consider the effects of temporal delay in a neural feedback system with excitation and inhibition. The topology of our model system reflects the anatomy of the avian isthmic circuitry, a feedback structure found in all classes of vertebrates. We show that the system is capable of performing a 'winner-take-all' selection rule for certain combinations of excitatory and inhibitory feedback. In particular, we show that when the time delays are sufficiently large a system with local inhibition and global excitation can function as a 'winner-take-all' network and exhibit oscillatory dynamics. We demonstrate how the origin of the oscillations can be attributed to the finite delays through a linear stability analysis.     

Modeling time series of animal behavior by means of a latent-state model with feedback

Summary .   We describe a family of models developed for time series of animal feeding behavior. The models incorporate both an unobserved state, which can be interpreted as the motivational state of the animal, and a mechanism for feedback to this state from the observed behavior. We discuss methods for evaluating and maximizing the likelihood of an observed series of behaviors, and thereby estimating parameters, and for inferring the most likely sequence of underlying states. We indicate several extensions of the models, including the incorporation of random effects. We apply these methods in an analysis of the feeding behavior of the caterpillar Helicoverpa armigera , and thereby demonstrate the potential of this family of models as a tool in the investigation of behavior.     

Nonlinear EEG analysis based on a neural mass model

The well-known neural mass model described by Lopes da Silva et al. (1976) and Zetterberg et al. (1978) is fitted to actual EEG data. This is achieved by reformulating the original set of integral equations as a continuous-discrete state space model. The local linearization approach is then used to discretize the state equation and to construct a nonlinear Kalman filter. On this basis, a maximum likelihood procedure is used for estimating the model parameters for several EEG recordings. The analysis of the noise-free differential equations of the estimated models suggests that there are two different types of alpha rhythms: those with a point attractor and others with a limit cycle attractor. These attractors are also found by means of a nonlinear time series analysis of the EEG recordings. We conclude that the Hopf bifurcation described by Zetterberg et al. (1978) is present in actual brain dynamics. Received: 11 August 1997 / Accepted in revised form: 20 April 1999     

Determining a distributed parameter in a neural cable model via a boundary control method

Dendrites of nerve cells have membranes with spatially distributed densities of ionic channels and hence non-uniform conductances. These conductances are usually represented as constant parameters in neural models because of the difficulty in experimentally estimating them locally. In this paper we investigate the inverse problem of recovering a single spatially distributed conductance parameter in a one-dimensional diffusion (cable) equation through a new use of a boundary control method. We also outline how our methodology can be extended to cable theory on finite tree graphs. The reconstruction is unique.     

Parameter estimation in a stochastic model of the tubuloglomerular feedback mechanism in a rat nephron

A key parameter in the understanding of renal hemodynamics is the gain of the feedback function in the tubuloglomerular feedback mechanism. A dynamic model of autoregulation of renal blood flow and glomerular filtration rate has been extended to include a stochastic differential equations model of one of the main parameters that determines feedback gain. The model reproduces fluctuations and irregularities in the tubular pressure oscillations that the former deterministic models failed to describe. This approach assumes that the gain exhibits spontaneous erratic variations that can be explained by a variety of influences, which change over time (blood pressure, hormone levels, etc.). To estimate the key parameters of the model we have developed a new estimation method based on the oscillatory behavior of the data. The dynamics is characterized by the spectral density, which has been estimated for the observed time series, and numerically approximated for the model. The parameters have then been estimated by the least squares distance between data and model spectral densities. To evaluate the estimation procedure measurements of the proximal tubular pressure from 35 nephrons in 16 rat kidneys have been analyzed, and the parameters characterizing the gain and the delay have been estimated. There was good agreement between the estimated values, and the values obtained for the same parameters in independent, previously published experiments.     

A model of the neural mechanisms responsible for pattern recognition and stimulus specific habituation in toads

A neural model of the mechanisms possibly responsible for stimulus-specific habituation in toads is proposed. The model follows the hypothesis that preypredator recognition is performed by command units as a result of retina-tectum-pretectum interaction. The model allow us to study the possible coding that the nervous system of toads uses for different prey stimuli, the neural mechanisms of habituation and dishabituation, and the dynamic changes that the command units may have during these processes. The model proposes specific hypothesis and experiments to clarify the nature of these processes and to test the validity of the command unit hypothesis.Research supported in part by CONACYT under grant PCCBBNA 021005Research supported in part by the NIH under grant NS 14971-05 from National Institute of Neurological and Communicative Disorders and Stroke     

Exponential synchronization of discontinuous neural networks with time-varying mixed delays via state feedback and impulsive control

This paper investigates drive-response synchronization for a class of neural networks with time-varying discrete and distributed delays (mixed delays) as well as discontinuous activations. Strict mathematical proof shows the global existence of Filippov solutions to neural networks with discontinuous activation functions and the mixed delays. State feedback controller and impulsive controller are designed respectively to guarantee global exponential synchronization of the neural networks. By using Lyapunov function and new analysis techniques, several new synchronization criteria are obtained. Moreover, lower bound on the convergence rate is explicitly estimated when state feedback controller is utilized. Results of this paper are new and some existing ones are extended and improved. Finally, numerical simulations are given to verify the effectiveness of the theoretical results.     

Modeling genetic switches with positive feedback loops

In this paper, we develop a new methodology to design synthetic genetic switch networks with multiple genes and time delays, by using monotone dynamical systems. We show that the networks with only positive feedback loops have no stable oscillation but stable equilibria whose stability is independent of the time delays. In other words, such systems have ideal properties for switch networks and can be designed without consideration of time delays, because the systems can be reduced from functional spaces to Euclidian spaces. Therefore, we can ensure that the designed switches function correctly even with uncertain delays. We first prove the basic properties of the genetic networks composed of only positive feedback loops, and then propose a procedure to design the switches, which drastically simplifies analysis of the switches and makes theoretical analysis and design tractable even for large-scaled systems. Finally, to demonstrate our theoretical results, we show biologically plausible examples by designing a synthetic genetic switch with experimentally well investigated lacI, tetR, and cI genes for numerical simulation.     

UKF-based closed loop iterative learning control of epileptiform wave in a neural mass model

A novel closed loop control framework is proposed to inhibit epileptiform wave in a neural mass model by external electric field, where the unscented Kalman filter method is used to reconstruct dynamics and estimate unmeasurable parameters of the model. Specifically speaking, the iterative learning control algorithm is introduced into the framework to optimize the control signal. In the proposed method, the control effect can be significantly improved based on the observation of the past attempts. Accordingly, the proposed method can effectively suppress the epileptiform wave as well as showing robustness to noises and uncertainties. Lastly, the simulation is carried out to illustrate the feasibility of the proposed method. Besides, this work shows potential value to design model-based feedback controllers for epilepsy treatment.     

A neural mass model based on single cell dynamics to model pathophysiology

Neural mass models are successful in modeling brain rhythms as observed in macroscopic measurements such as the electroencephalogram (EEG). While the synaptic current is explicitly modeled in current models, the single cell electrophysiology is not taken into account. To allow for investigations of the effects of channel pathologies, channel blockers and ion concentrations on macroscopic activity, we formulate neural mass equations explicitly incorporating the single cell dynamics by using a bottom-up approach. The mean and variance of the firing rate and synaptic input distributions are modeled. The firing rate curve (F(I)-curve) is used as link between the single cell and macroscopic dynamics. We show that this model accurately reproduces the behavior of two populations of synaptically connected Hodgkin-Huxley neurons, also in non-steady state.     

A rat model for neural circuitry abnormalities in schizophrenia

Animal models for complex brain disorders, such as schizophrenia, are essential for the interpretation of postmortem findings. These models allow empirical testing of hypotheses regarding the role of genetic and environmental factors, the pathophysiological mechanisms and brain circuits that are responsible for specific neural abnormalities and their associated behavioral impairment, and the effectiveness of therapeutic treatments relative to these diseases. Recently, we developed a rodent model for neural circuitry abnormalities in discrete corticolimbic subregions of subjects with major psychoses. According to our protocol, the GABA-A receptor antagonist picrotoxin is stereotaxically infused in the basolateral amygdala to mimic a GABA defect in this region that is postulated to occur in these disorders. This protocol has been tested with a number of acute and chronic time schedules. Following picrotoxin administration in the basolateral amygdala, changes in GABAergic neurons and/or terminals in hippocampal regions CA2/3 are observed, similar to those seen in major psychoses, as well as a marked reduction in GABA-receptor-mediated currents in pyramidal neurons of this region. This has established the construct and predictive validity of this model for studying limbic-lobe circuitry abnormalities. We propose that this modeling strategy may provide a valid alternative to isomorphic models of these diseases.     

Long-term retention of behavioural habituation in the rat

The orienting response in rats was measured by the interruption in licking that occurred during the presentation of a tone stimulus. Habituation occurred after a mean of 6·3 presentations and there was complete retention of habituation for 72 hr, and 70 per cent retention after 288 hr. This long-term storage involved in habituation distinguishes it from fatigue and supports the view that it must be regarded as a type of learning.