Estimating causal interaction between prefrontal cortex and striatum by transfer entropy

Transfer entropy (TE) is an information-theoretic measure for the investigation of causal interaction between two systems without a requirement of pre-specific interaction model (such as: linear or nonlinear). We introduced an efficient algorithm to calculate TE values between two systems based on observed time signals. By this method, we demonstrated that the TE correctly estimated the coupling strength and the direction of information transmission of two nonlinearly coupled systems. We also calculated TE values of real local field potentials (LFPs) recorded simultaneously in the lateral prefrontal cortex (LPFC) and the striatum of the behavioral monkey, and observed that the TE value from the LPFC to the striatum was stronger than that from the striatum to the LPFC, consistent with anatomical structure between the two areas. Moreover, the TE value dynamically varied dependent on behavior stages of the monkey. These results from simulated and real LFPs data suggested that the TE was able to effectively estimate functional connectivity between different brain regions and characterized their dynamical properties.     

A coupled oscillator model of disordered interlimb coordination in patients with Parkinson's disease

Coordination between the left and right limbs during cyclic movements, which can be characterized by the amplitude of each limb's oscillatory movement and relative phase, is impaired in patients with Parkinson's disease (PD). A pedaling exercise on an ergometer in a recent clinical study revealed several types of coordination disorder in PD patients. These include an irregular and burst-like amplitude modulation with intermittent changes in its relative phase, a typical sign of chaotic behavior in nonlinear dynamical systems. This clinical observation leads us to hypothesize that emergence of the rhythmic motor behaviors might be concerned with nonlinearity of an underlying dynamical system. In order to gain insight into this hypothesis, we consider a simple hard-wired central pattern generator model consisting of two identical oscillators connected by reciprocal inhibition. In the model, each oscillator acts as a neural half-center controlling movement of a single limb, either left or right, and receives a control input modeling a flow of descending signals from higher motor centers. When these two control inputs are tonic-constant and identical, the model has left-right symmetry and basically exhibits ordered coordination with an alternating periodic oscillation. We show that, depending on the intensities of these two control inputs and on the difference between them that introduces asymmetry into the model, the model can reproduce several behaviors observed in the clinical study. Bifurcation analysis of the model clarifies two possible mechanisms for the generation of disordered coordination in the model: one is the spontaneous symmetry-breaking bifurcation in the model with the left-right symmetry. The other is related to the degree of asymmetry reflecting the difference between the two control inputs. Finally, clinical implications by the model's dynamics are briefly discussed.     

Phase-Coherence Transitions and Communication in the Gamma Range between Delay-Coupled Neuronal Populations

Synchronization between neuronal populations plays an important role in information transmission between brain areas. In particular, collective oscillations emerging from the synchronized activity of thousands of neurons can increase the functional connectivity between neural assemblies by coherently coordinating their phases. This synchrony of neuronal activity can take place within a cortical patch or between different cortical regions. While short-range interactions between neurons involve just a few milliseconds, communication through long-range projections between different regions could take up to tens of milliseconds. How these heterogeneous transmission delays affect communication between neuronal populations is not well known. To address this question, we have studied the dynamics of two bidirectionally delayed-coupled neuronal populations using conductance-based spiking models, examining how different synaptic delays give rise to in-phase/anti-phase transitions at particular frequencies within the gamma range, and how this behavior is related to the phase coherence between the two populations at different frequencies. We have used spectral analysis and information theory to quantify the information exchanged between the two networks. For different transmission delays between the two coupled populations, we analyze how the local field potential and multi-unit activity calculated from one population convey information in response to a set of external inputs applied to the other population. The results confirm that zero-lag synchronization maximizes information transmission, although out-of-phase synchronization allows for efficient communication provided the coupling delay, the phase lag between the populations, and the frequency of the oscillations are properly matched.     

Mapping information flow in sensorimotor networks

Biological organisms continuously select and sample information used by their neural structures for perception and action, and for creating coherent cognitive states guiding their autonomous behavior. Information processing, however, is not solely an internal function of the nervous system. Here we show, instead, how sensorimotor interaction and body morphology can induce statistical regularities and information structure in sensory inputs and within the neural control architecture, and how the flow of information between sensors, neural units, and effectors is actively shaped by the interaction with the environment. We analyze sensory and motor data collected from real and simulated robots and reveal the presence of information structure and directed information flow induced by dynamically coupled sensorimotor activity, including effects of motor outputs on sensory inputs. We find that information structure and information flow in sensorimotor networks (a) is spatially and temporally specific; (b) can be affected by learning, and (c) can be affected by changes in body morphology. Our results suggest a fundamental link between physical embeddedness and information, highlighting the effects of embodied interactions on internal (neural) information processing, and illuminating the role of various system components on the generation of behavior.     

Physics of directional hearing in the cricket Gryllus bimaculatus

In the cricket ear, sound acts on the external surface of the tympanum and also reaches the inner surface after travelling in at least three pathways in the tracheal system. We have determined the transmission gain of the three internal sound pathways; that is, the change of amplitude and phase angle from the entrances of the tracheal system to the inner surface of the tympanum. In addition, we have measured the diffraction and time of arrival of sound at the ear and at the three entrances at various directions of sound incidence. By combining these data we have calculated how the total driving force at the tympanum depends on the direction of sound. The results are in reasonable agreement with the directionality of the tympanal vibrations as determined with laser vibrometry.At the frequency of the calling song (4.7 kHz), the direction of the sound has little effect on the amplitudes of the sounds acting on the tympanum, but large effects on their phase angles, especially of the sound waves entering the tracheal system at the contralateral side of the body. The master parameter for causing the directionality of the ear in the forward direction is the sound wave entering the contralateral thoracic spiracle. The phase of this sound component may change by 130–140° with sound direction. The transmission of sound from the contralateral inputs is dominated by a very selective high-pass filter, and large changes in amplitude and phase are seen in the transmitted sounds when the sound frequency changes from 4 to 5 kHz. The directionality is therefore very dependent on sound frequency.The transmission gains vary considerably in different individuals, and much variation was also found in the directional patterns of the ears, especially in the effects of sounds from contralateral directions. However, the directional pattern in the frontal direction is quite robust (at least 5 dB difference between the 330° and 30° directions), so these variations have only little effect on how well the individual animals can approach singing conspecifics.Abbreviations CS contralateral spiracle - CT contralateral tympanum - IS ipsilateral spiracle - IT ipsilateral tympanum - P the vectorial sum of the sounds acting on the tympanum     

Analysis of the mechanisms of interaction between generators of cyclic motor reactions and inputs from suprasegmental systems

The simulation mathematical model of neuronal generator systems was used for analyzing the interaction between inputs from descending (afferent) systems and generators of scratching (locomotion). The data obtained indicate that in cases when generators of cyclic motor reactions influence the effectiveness of synaptic transmission from the fibers of descending (or afferent) systems, the differences in the main characteristics of signals that are produced by these generators themselves and those that come to them are emphasized. These data allow us to conclude that the system supplying interaction between inputs from suprasegmental (afferent) fiber systems and generators of locomotion of scratching can be interpreted as an adaptive filter which processes spatial and temporal information coming to the spinal cord via different suprasegmental or primary afferent inputs and allows generators of cyclic motor reactions to correct their functioning in accordance with changing external conditions.Neirofiziologiya/Neurophysiology, Vol. 25, No. 3, pp. 211–215, May–June, 1993.     

Analysis of chaotic oscillations induced in two coupled Wilson–Cowan models

Although it is known that two coupled Wilson–Cowan models with reciprocal connections induce aperiodic oscillations, little attention has been paid to the dynamical mechanism for such oscillations so far. In this study, we aim to elucidate the fundamental mechanism to induce the aperiodic oscillations in the coupled model. First, aperiodic oscillations observed are investigated for the case when the connections are unidirectional and when the input signal is a periodic oscillation. By the phase portrait analysis, we determine that the aperiodic oscillations are caused by periodically forced state transitions between a stable equilibrium and a stable limit cycle attractors around the saddle-node and saddle separatrix loop bifurcation points. It is revealed that the dynamical mechanism where the state crosses over the saddle-node and saddle separatrix loop bifurcations significantly contributes to the occurrence of chaotic oscillations forced by a periodic input. In addition, this mechanism can also give rise to chaotic oscillations in reciprocally connected Wilson–Cowan models. These results suggest that the dynamic attractor transition underlies chaotic behaviors in two coupled Wilson–Cowan oscillators.     

Computational models of signalling networks for non-linear control

Artificial signalling networks (ASNs) are a computational approach inspired by the signalling processes inside cells that decode outside environmental information. Using evolutionary algorithms to induce complex behaviours, we show how chaotic dynamics in a conservative dynamical system can be controlled. Such dynamics are of particular interest as they mimic the inherent complexity of non-linear physical systems in the real world. Considering the main biological interpretations of cellular signalling, in which complex behaviours and robust cellular responses emerge from the interaction of multiple pathways, we introduce two ASN representations: a stand-alone ASN and a coupled ASN. In particular we note how sophisticated cellular communication mechanisms can lead to effective controllers, where complicated problems can be divided into smaller and independent tasks.     

Two coupled oscillators as a model for the coordinated finger tapping by both hands

Recently, it was found that rhythmic movements (e.g. locomotion, swimmeret beating) are controlled by mutually coupled endogeneous neural oscillators (Kennedy and Davis, 1977; Pearson and Iles, 1973; Stein, 1974; Shik and Orlovsky, 1976; Grillner and Zangger, 1979). Meanwhile, it has been found out that the phase resetting experiment is useful to investigate the interaction of neural oscillators (Perkel et al., 1963; Stein, 1974). In the preceding paper (Yamanishi et al., 1979), we studied the functional interaction between the neural oscillator which is assumed to control finger tapping and the neural networks which control some tasks. The tasks were imposed on the subject as the perturbation of the phase resetting experiment. In this paper, we investigate the control mechanism of the coordinated finger tapping by both hands. First, the subjects were instructed to coordinate the finger tapping by both hands so as to keep the phase difference between two hands constant. The performance was evaluated by a systematic error and a standard deviation of phase differences. Second, we propose two coupled neural oscillators as a model for the coordinated finger tapping. Dynamical behavior of the model system is analyzed by using phase transition curves which were measured on one hand finger tapping in the previous experiment (Yamanishi et al., 1979). Prediction by the model is in good agreement with the results of the experiments. Therefore, it is suggested that the neural mechanism which controls the coordinated finger tapping may be composed of a coupled system of two neural oscillators each of which controls the right and the left finger tapping respectively.     

Mutual Information Rate and Bounds for It

The amount of information exchanged per unit of time between two nodes in a dynamical network or between two data sets is a powerful concept for analysing complex systems. This quantity, known as the mutual information rate (MIR), is calculated from the mutual information, which is rigorously defined only for random systems. Moreover, the definition of mutual information is based on probabilities of significant events. This work offers a simple alternative way to calculate the MIR in dynamical (deterministic) networks or between two time series (not fully deterministic), and to calculate its upper and lower bounds without having to calculate probabilities, but rather in terms of well known and well defined quantities in dynamical systems. As possible applications of our bounds, we study the relationship between synchronisation and the exchange of information in a system of two coupled maps and in experimental networks of coupled oscillators.     

Synchrony &; chaos in patchy ecosystems

The apparent synchronisation of spatially discrete populations is a well documented phenomenon. However, it is not clear what the governing mechanisms are for this synchrony, and whether they are robust over a range of environmental conditions and patch specific population dynamic behaviours. In this paper, we explore two (possibly interacting) modes of coupling, and investigate their theoretically discernible, and perhaps even experimentally measurable, signatures. To aid us in this investigation we employ a planktonic example system, with direct application to plankton patchiness. Furthermore, we address the role of chaos in complex spatio-temporal dynamics; we find that chaos associated with funnel attractors can play a distinguished role, over dynamics less sensitive to small variations, in being more susceptible to generalised synchronisation (such as phase synchronisation) in the presence of small local parameter variation. This is in contrast to the case for coupled systems with identical dynamics, and suggests that non-identically coupled systems are more vulnerable to global extinction events when exhibiting funnel-type chaotic dynamics.     

Biochemical switching device: how to turn on (off) the switch

We previously showed with computer simulations that cyclic enzyme systems have the reliability of ON-OFF types of operation (McCulloch-Pitts' neuronic equation) and the applicability for a switching circuit in a biocomputer. The switching time was inevitably determined in accordance with the difference in amount between two inputs of the system. This characteristic is, however, a disadvantage for practical use of a switching device; we need to improve the system in order for the switching time to optionally be changed. We shall present here how to turn on (off) the switch independently of the modes of two inputs. By introducing pulse perturbation, we could optionally set up the switching time of a cyclic enzyme system (biochemical switching device).     

Frequency dependence of the action-perception cycle for postural control in a moving visual environment: relative phase dynamics

When standing human subjects are exposed to a moving visual environment, the induced postural sway displays varying degrees of coherence with the visual information. In our experiment we varied the frequency of an oscillatory visual display and analysed the temporal relationship between visual motion and sway. We found that subjects maintain sizeable sway amplitudes even as temporal coherence with the display is lost. Postural sway tended to phase lead (for frequencies below 0.2 Hz) or phase lag (above 0.3 Hz). However, we also observed at a fixed frequency, highly variable phase relationships in which a preferred range of phase lags is prevalent, but phase jumps occur that return the system into the preferred range after phase has begun drifting out of the preferred regime. By comparing the results quantitatively with a dynamical model (the sine-circle map), we show that this effect can be understood as a form of relative coordination and arises through an instability of the dynamics of the action-perception cycle. Because such instabilities cannot arise in passively driven systems, we conclude that postural sway in this situation is actively generated as rhythmic movement which is coupled dynamically to the visual motion. Received: 7 September 1993/Accepted in revised form: 2 May 1994     

Why do evolutionary systems stick to the edge of chaos

Summary  The long-term behaviour of dynamic systems can be classified in two different regimes, regular or chaotic, depending on the values of the control parameters, which are kept constant during the time evolution. Starting from slightly different initial conditions, a regular system converges to the same final trajectory, whereas a chaotic system follows two distinct trajectories exponentially diverging from each other. In spite of these differences, regular and chaotic systems share a common property: both arrive exponentially fast to their final destiny, becoming trapped there. In both cases one has finite transient times. This is not a profitable property in what concerns evolutionary strategies, where the eternal search for new forms, better than the current one, is imperative. That is why evolutionary dynamic systems tend to tune themselves in very particular situations in between regular and chaotic regimes. These particular situations present eternal transients, and the system actually never reaches its final destiny, preserving diversity. This feature allows the system to visit other regions of the space of possibilities, not only the tiny region covered by its final attractor.     

How simple grazing rules can lead to persistent boundaries in vegetation communities

Natural landscape boundaries between vegetation communities are dynamically influenced by the selective grazing of herbivores. Here we show how this may be an emergent property of very simple animal decisions, without the need for any sophisticated choice rules etc., using a model based on biased diffusion. Animal grazing intensity is coupled with plant competition, resulting in reaction-diffusion dynamics, from which stable boundaries spontaneously emerge. In the model, animals affect their resources by both consumption and trampling. It is assumed that forage consists of two heterogeneously distributed competing resource species, one that is preferred (grass) over the other (heather) by the animals. The solutions to the resulting system of differential equations for three cases a) optimal foraging, b) random walk foraging and c) taxis-diffusion are presented. Optimal and random foraging gave unrealistic results, but taxis-diffusion accorded well with field observations. Persistent boundaries between patches of near-monoculture vegetation were predicted, with these boundaries drifting in response to overall grazing pressure (grass advancing with increased grazing and vice versa). The reaction-taxis-diffusion model provides the first mathematical explanation for such vegetation mosaic dynamics and the parameters of the model are open to experimental testing.     

The Frequency Response, Coherence, and Information Capacity of Two Neuronal Models

Two neuronal models are analyzed in which subthreshold inputs are integrated either without loss (perfect integrator) or with a decay which follows an exponential time course (leaky integrator). Linear frequency response functions for these models are compared using sinusoids, Poisson-distributed impulses, or gaussian white noise as inputs. The responses of both models show the nonlinear behavior characteristic of a rectifier for sinusoidal inputs of sufficient amplitude. The leaky integrator shows another nonlinearity in which responses become phase locked to cyclic stimuli. Addition of white noise reduces the distortions due to phase locking. Both models also show selective attenuation of high-frequency components with white noise inputs. Input, output, and cross-spectra are computed using inputs having a broad frequency spectrum. Measures of the coherence and information transmission between the input and output of the models are also derived. Steady inputs, which produce a constant “carrier” rate, and intrinsic sources, which produce variability in the discharge of neurons, may either increase or decrease coherence; however, information transmission using inputs with a broad spectrum is generally increased by steady inputs and reduced by intrinsic variability.     

Synaptic dendritic activity modulates the single synaptic event

Synaptic transmission is the key system for the information transfer and elaboration among neurons. Nevertheless, a synapse is not a standing alone structure but it is a part of a population of synapses inputting the information from several neurons on a specific area of the dendritic tree of a single neuron. This population consists of excitatory and inhibitory synapses the inputs of which drive the postsynaptic membrane potential in the depolarizing (excitatory synapses) or depolarizing (inhibitory synapses) direction modulating in such a way the postsynaptic membrane potential. The postsynaptic response of a single synapse depends on several biophysical factors the most important of which is the value of the membrane potential at which the response occurs. The concurrence in a specific time window of inputs by several synapses located in a specific area of the dendritic tree can, consequently, modulate the membrane potential such to severely influence the single postsynaptic response. The degree of modulation operated by the synaptic population depends on the number of synapses active, on the relative proportion between excitatory and inbibitory synapses belonging to the population and on their specific mean firing frequencies. In the present paper we show results obtained by the simulation of the activity of a single Glutamatergic excitatory synapse under the influence of two different populations composed of the same proportion of excitatory and inhibitory synapses but having two different sizes (total number of synapses). The most relevant conclusion of the present simulations is that the information transferred by the single synapse is not and independent simple transition between a pre- and a postsynaptic neuron but is the result of the cooperation of all the synapses which concurrently try to transfer the information to the postsynaptic neuron in a given time window. This cooperativeness is mainly operated by a simple mechanism of modulation of the postsynaptic membrane potential which influences the amplitude of the different components forming the postsynaptic excitatory response.     

Gamma oscillations as a mechanism for selective information transmission

In the past decades, many studies have focussed on the relation between the input and output of neurons with the aim to understand information processing by neurons. A particular aspect of neuronal information, which has not received much attention so far, concerns the problem of information transfer when a neuron or a population of neurons receives input from two or more (populations of) neurons, in particular when these (populations of) neurons carry different types of information. The aim of the present study is to investigate the responses of neurons to multiple inputs modulated in the gamma frequency range. By a combination of theoretical approaches and computer simulations, we test the hypothesis that enhanced modulation of synchronized excitatory neuronal activity in the gamma frequency range provides an advantage over a less synchronized input for various types of neurons. The results of this study show that the spike output of various types of neurons [i.e. the leaky integrate and fire neuron, the quadratic integrate and fire neuron and the Hodgkin–Huxley (HH) neuron] and that of excitatory–inhibitory coupled pairs of neurons, like the Pyramidal Interneuronal Network Gamma (PING) model, is highly phase-locked to the larger of two gamma-modulated input signals. This implies that the neuron selectively responds to the input with the larger gamma modulation if the amplitude of the gamma modulation exceeds that of the other signals by a certain amount. In that case, the output of the neuron is entrained by one of multiple inputs and that other inputs are not represented in the output. This mechanism for selective information transmission is enhanced for short membrane time constants of the neuron.     

Characteristics and plasticity of electrical synaptic transmission

Electrical synapses are an omnipresent feature of nervous systems, from the simple nerve nets of cnidarians to complex brains of mammals. Formed by gap junction channels between neurons, electrical synapses allow direct transmission of voltage signals between coupled cells. The relative simplicity of this arrangement belies the sophistication of these synapses. Coupling via electrical synapses can be regulated by a variety of mechanisms on times scales ranging from milliseconds to days, and active properties of the coupled neurons can impart emergent properties such as signal amplification, phase shifts and frequency-selective transmission. This article reviews the biophysical characteristics of electrical synapses and some of the core mechanisms that control their plasticity in the vertebrate central nervous system.     

Coupled predator–prey oscillations in a chaotic food web

Coupling of several predator–prey oscillations can generate intriguing patterns of synchronization and chaos. Theory predicts that prey species will fluctuate in phase if predator–prey cycles are coupled through generalist predators, whereas they will fluctuate in anti-phase if predator–prey cycles are coupled through competition between prey species. Here, we investigate predator–prey oscillations in a long-term experiment with a marine plankton community. Wavelet analysis of the species fluctuations reveals two predator–prey cycles that fluctuate largely in anti-phase. The phase angles point at strong competition between the phytoplankton species, but relatively little prey overlap among the zooplankton species. This food web architecture is consistent with the size structure of the plankton community, and generates highly dynamic food webs. Continued alternations in species dominance enable coexistence of the prey species through a non-equilibrium 'killing-the-winner' mechanism, as the system shifts back and forth between the two predator–prey cycles in a chaotic fashion.