Dynamics and bifurcations of two coupled neural oscillators with different connection types

In this paper we present an oscillatory neural network composed of two coupled neural oscillators of the Wilson-Cowan type. Each of the oscillators describes the dynamics of average activities of excitatory and inhibitory populations of neurons. The network serves as a model for several possible network architectures. We study how the type and the strength of the connections between the oscillators affect the dynamics of the neural network. We investigate, separately from each other, four possible connection types (excitatory→excitatory, excitatory→inhibitory, inhibitory→excitatory, and inhibitory→inhibitory) and compute the corresponding bifurcation diagrams. In case of weak connections (small strength), the connection of populations of different types lead to periodicin-phase oscillations, while the connection of populations of the same type lead to periodicanti-phase oscillations. For intermediate connection strengths, the networks can enter quasiperiodic or chaotic regimes, and can also exhibit multistability. More generally, our analysis highlights the great diversity of the response of neural networks to a change of the connection strength, for different connection architectures. In the discussion, we address in particular the problem of information coding in the brain using quasiperiodic and chaotic oscillations. In modeling low levels of information processing, we propose that feature binding should be sought as a temporally coherent phase-locking of neural activity. This phase-locking is provided by one or more interacting convergent zones and does not require a central “top level” subcortical circuit (e.g. the septo-hippocampal system). We build a two layer model to show that although the application of a complex stimulus usually leads to different convergent zones with high frequency oscillations, it is nevertheless possible to synchronize these oscillations at a lower frequency level using envelope oscillations. This is interpreted as a feature binding of a complex stimulus.     

Morphological characterization of single fan-shaped body neurons in <Emphasis Type="Italic">Drosophila melanogaster</Emphasis>

The fan-shaped body is the largest substructure of the central complex in Drosophila melanogaster. Two groups of large-field neurons that innervate the fan-shaped body, viz., F1 and F5 neurons, have recently been found to be involved in visual pattern memory for “contour orientation” and “elevation” in a rut-dependent manner. The F5 neurons have been found to be responsible for the parameter “elevation” in a for-dependent manner. We have shown here that the F1 neuron also affects visual memory for “contour orientation” in a for-dependent way. With the help of Gal4/UAS and FLP-out techniques, we have characterized the morphological features of these two groups of neurons at single neuron resolution. We have observed that F1 or F5 neurons are groups of isomorphic individual neurons. Single F1 neurons have three main arborization regions: one in the first layer of the fan-shaped body, one in the ventral body, and another in the inferior medial protocerebrum. Single F5 neurons have two arborization regions: one in the fifth layer of the fan-shaped body and the other in the superior medial protocerebrum. The polarity of the F1 and F5 neurons has been studied with the Syt-GFP marker. Our results indicate the existence of presynaptic sites of both F1 and F5 neurons located in the fan-shaped body and postsynaptic sites outside of the fan-shaped body. This work was supported by the “973 Program” (2005CB522804 and 2009CB918702), the National Natural Sciences Foundation of China (30621004, 30625022, and 30770682), and the Knowledge Innovation Program of the Chinese Academy of Sciences (KSCX2-YW-R-28).     

Simple models for excitable and oscillatory neural networks

 Chains of coupled oscillators of simple “rotator” type have been used to model the central pattern generator (CPG) for locomotion in lamprey, among numerous applications in biology and elsewhere. In this paper, motivated by experiments on lamprey CPG with brainstem attached, we investigate a simple oscillator model with internal structure which captures both excitable and bursting dynamics. This model, and that for the coupling functions, is inspired by the Hodgkin–Huxley equations and two-variable simplifications thereof. We analyse pairs of coupled oscillators with both excitatory and inhibitory coupling. We also study traveling wave patterns arising from chains of oscillators, including simulations of “body shapes” generated by a double chain of oscillators providing input to a kinematic musculature model of lamprey.. Received: 25 November 1996 / Revised version: 9 December 1997     

Slow modulation of synaptic transmission by brain-derived neurotrophic factor leads to the central sensitization associated with neuropathic pain

Chronic constriction injury (CCI) of the rat sciatic nerve increases the dorsal horn excitability. This “central sensitization” leads to behavioral manifestations analogous to those related to human neuropathic pain. We found, using whole-cell recording from acutely isolated spinal cord slices, that 7-to 10-day-long CCI increases excitatory synaptic drive to putative excitatory “delay”-firing neurons in the substantia gelatinosa but attenuates that to putative inhibitory “tonic”-firing neurons. A defined-medium organotypic culture (DMOTC) system was used to investigate the long-term actions of brain-derived neurotrophic factor (BDNF) as a possible instigator of these changes. When all five neuronal types found in the substantia gelatinosa were considered, BDNF and CCI produced similar patterns, or “footprints,” of changes across the whole population. This pattern was not seen with another putative “pain mediator,” interleukin 1β. Thus, BDNF decreased synaptic drive to “tonic” neurons and increased synaptic drive to “delay” neurons. Actions of BDNF on “delay” neurons were presynaptic and involved increased mEPSC frequency and amplitude without changes in the function of postsynaptic AMPA receptors. By contrast, BDNF exerted both pre-and post-synaptic actions on “ tonic” cells to reduce the mEPSC frequency and amplitude. These differential actions of BDNF on excitatory and inhibitory neurons contributed to a global increase in the dorsal horn network excitability as assessed by the amplitude of depolarization-induced increases in the intracellular [Ca²⁺]. Experiments with the BDNF-binding protein TrkB-d5 provided additional evidence for BDNF as a harbinger of neuropathic pain. Thus, the cellular processes altered by BDNF likely contribute to “central sensitization” and hence to the onset of neuropathic pain. Neirofiziologiya/Neurophysiology, Vol. 39, Nos. 4/5, pp. 315–326, July–October, 2007.     

Background and reflex activity of guinea pig caudal mesenteric ganglion neurons

The background activity of the guinea pig caudal mesenteric ganglion (CMG) neurons and their reflex reactions to colonic distension were studied on isolated combined preparations including the CMG and a colon segment connected with the lumbar colonic nerves. In the control, 62% of the neurons under study generated background activity, which consisted of irregular or regular “fast” excitatory postsynaptic potentials (fEPSP) and action potentials (AP). In 27% of the CMG neurons called “pacemaker-like neurons” (PLN), the background activity was represented by highly regular AP never observed in the CMG completely isolated from the distal colon. Reflex responses evoked by colonic distension were recorded from 76% of the units studied. The distension evoked fEPSP and AP in “silent” neurons and increased the background activity. Both the background activity and reflex responses were shown to be due to nicotinic cholinergic transmission. In some neurons, reflex responses (regular AP) were generated as superimposed on a slow depolarization; the latter was insensitive to nicotinic antagonists and either sensitive or insensitive to muscarinic antagonists. It has been concluded that CMG neurons receive nicotinic, muscarinic, and, probably, peptidergic afferent inputs from the distal colon. Although there are no true pacemaker neurons in CMG, some neurons generate pacemaker-like activity of a synaptic origin.     

Cortical Resonance Frequencies Emerge from Network Size and Connectivity

Neural oscillations occur within a wide frequency range with different brain regions exhibiting resonance-like characteristics at specific points in the spectrum. At the microscopic scale, single neurons possess intrinsic oscillatory properties, such that is not yet known whether cortical resonance is consequential to neural oscillations or an emergent property of the networks that interconnect them. Using a network model of loosely-coupled Wilson-Cowan oscillators to simulate a patch of cortical sheet, we demonstrate that the size of the activated network is inversely related to its resonance frequency. Further analysis of the parameter space indicated that the number of excitatory and inhibitory connections, as well as the average transmission delay between units, determined the resonance frequency. The model predicted that if an activated network within the visual cortex increased in size, the resonance frequency of the network would decrease. We tested this prediction experimentally using the steady-state visual evoked potential where we stimulated the visual cortex with different size stimuli at a range of driving frequencies. We demonstrate that the frequency corresponding to peak steady-state response inversely correlated with the size of the network. We conclude that although individual neurons possess resonance properties, oscillatory activity at the macroscopic level is strongly influenced by network interactions, and that the steady-state response can be used to investigate functional networks.     

Perception of spatial relations between objects in early ontogeny

Development of the perception of spatial relations between objects was studied in infants aged from 3 to 4 to 24 to 25 months. The following tests were performed: prediction of the results of rectilinear and nonrectilinear toy motion; search for the toy hidden before the baby’s eyes under a cup, under one of two to five similar cups, and under a cup different from the others (a “local mark”), which was stationary or moving in the visual field; and search for a toy hidden under the “local mark” while distracting the baby's attention. It was shown that a child masters the regularities of spatial motion of an object first (at the age of 4 to 5 to 8 to 9 months). To the age of 10 to 11 months, all the children remember the location of a hidden toy using the egocentric location strategy (“Self” and “Object”). This strategy gradually improves and allows a child to ignore indifferent objects in the visual field. The ability to use a “local mark,” as a direct indicator of a hidden toy location, appears and is strengthened beginning from 14 to 15 months of age. This fact testifies to the transition from the egocentric strategy of object location to assessment of the relative location of two objects in the visual field. A capability for estimating the relative spatial position of three and more objects develops beginning from the age of two years.     

Schema design and implementation of the grasp-related mirror neuron system

 Mirror neurons within a monkey's premotor area F5 fire not only when the monkey performs a certain class of actions but also when the monkey observes another monkey (or the experimenter) perform a similar action. It has thus been argued that these neurons are crucial for understanding of actions by others. We offer the hand-state hypothesis as a new explanation of the evolution of this capability: the basic functionality of the F5 mirror system is to elaborate the appropriate feedback – what we call the hand state– for opposition-space based control of manual grasping of an object. Given this functionality, the social role of the F5 mirror system in understanding the actions of others may be seen as an exaptation gained by generalizing from one's own hand to an other's hand. In other words, mirror neurons first evolved to augment the “canonical” F5 neurons (active during self-movement based on observation of an object) by providing visual feedback on “hand state,” relating the shape of the hand to the shape of the object. We then introduce the MNS1 (mirror neuron system 1) model of F5 and related brain regions. The existing Fagg–Arbib–Rizzolatti–Sakata model represents circuitry for visually guided grasping of objects, linking the anterior intraparietal area (AIP) with F5 canonical neurons. The MNS1 model extends the AIP visual pathway by also modeling pathways, directed toward F5 mirror neurons, which match arm–hand trajectories to the affordances and location of a potential target object. We present the basic schemas for the MNS1 model, then aggregate them into three “grand schemas”– visual analysis of hand state, reach and grasp, and the core mirror circuit – for each of which we present a useful implementation (a non-neural visual processing system, a multijoint 3-D kinematics simulator, and a learning neural network, respectively). With this implementation we show how the mirror system may learnto recognize actions already in the repertoire of the F5 canonical neurons. We show that the connectivity pattern of mirror neuron circuitry can be established through training, and that the resultant network can exhibit a range of novel, physiologically interesting behaviors during the process of action recognition. We train the system on the basis of final grasp but then observe the whole time course of mirror neuron activity, yielding predictions for neurophysiological experiments under conditions of spatial perturbation, altered kinematics, and ambiguous grasp execution which highlight the importance of the timingof mirror neuron activity. Received: 6 August 2001 / Accepted in revised form: 5 February 2002     

A reinterpretation of the mathematical biophysics of the central nervous system in the light of neurophysiological findings

The fundamental equations for the interaction between neurons used in mathematical biophysics seem at first incompatible with the actual neurophysiological findings on the synaptic transmission. It is shown, however, that those equations may be readily interpreted in terms of accepted neurophysiological views. What has been termed “synapse” in mathematical biophysics must be regarded as a complicated network of internuncial neurons. It is shown that, under rather conditions, the number of those interneurons willstatistically vary with time according to the differential equation postulated for the excitatory and inhibitory factors. The latter are thus interpreted as the number of excitatory and inhibitory interneurons.     

Computational model for perception of objects and motions

Perception of objects and motions in the visual scene is one of the basic problems in the visual system. There exist ‘What’ and ‘Where’ pathways in the superior visual cortex, starting from the simple cells in the primary visual cortex. The former is able to perceive objects such as forms, color, and texture, and the latter perceives ‘where’, for example, velocity and direction of spatial movement of objects. This paper explores brain-like computational architectures of visual information processing. We propose a visual perceptual model and computational mechanism for training the perceptual model. The computational model is a three-layer network. The first layer is the input layer which is used to receive the stimuli from natural environments. The second layer is designed for representing the internal neural information. The connections between the first layer and the second layer, called the receptive fields of neurons, are self-adaptively learned based on principle of sparse neural representation. To this end, we introduce Kullback-Leibler divergence as the measure of independence between neural responses and derive the learning algorithm based on minimizing the cost function. The proposed algorithm is applied to train the basis functions, namely receptive fields, which are localized, oriented, and bandpassed. The resultant receptive fields of neurons in the second layer have the characteristics resembling that of simple cells in the primary visual cortex. Based on these basis functions, we further construct the third layer for perception of what and where in the superior visual cortex. The proposed model is able to perceive objects and their motions with a high accuracy and strong robustness against additive noise. Computer simulation results in the final section show the feasibility of the proposed perceptual model and high efficiency of the learning algorithm.     

Selective visual attention in a neurocomputational model of phase oscillators

In order to understand the dynamic property of covert selective visual attention, which is different from the proposed mechanism of the spotlight metaphor, a two-layered network of phase oscillators was developed. The first layer is related to the hippocampus and controls attention focus formation. The second layer is related to the visual cortex, and each cortical oscillator in it simulates an assembly of cells coding for a particular stimulus in the sense of feature binding. Selective visual attention is interpreted as the result of the emergent synchronization of hippocampus oscillators and a part of cortical oscillators. Numerical experiments are presented to illustrate attention focus formation and attention shifting from one set of stimuli to another. From a neurocomputational point of view, our results demonstrate that attention is an emergent property of the dynamical cell assemblies responding to the whole visual field. Received: 2 January 1998 / Accepted in revised form: 10 November 1998     

Neural field model of binocular rivalry waves

We present a neural field model of binocular rivalry waves in visual cortex. For each eye we consider a one-dimensional network of neurons that respond maximally to a particular feature of the corresponding image such as the orientation of a grating stimulus. Recurrent connections within each one-dimensional network are assumed to be excitatory, whereas connections between the two networks are inhibitory (cross-inhibition). Slow adaptation is incorporated into the model by taking the network connections to exhibit synaptic depression. We derive an analytical expression for the speed of a binocular rivalry wave as a function of various neurophysiological parameters, and show how properties of the wave are consistent with the wave-like propagation of perceptual dominance observed in recent psychophysical experiments. In addition to providing an analytical framework for studying binocular rivalry waves, we show how neural field methods provide insights into the mechanisms underlying the generation of the waves. In particular, we highlight the important role of slow adaptation in providing a “symmetry breaking mechanism” that allows waves to propagate.     

Structure-Dependent Electrical and Concentration Processes in the Dendrites of Pyramidal Neurons of Superficial Neocortical Layers: Model Study

On mathematical models of pyramidal neurons localized in the neocortical layers 2/3, whose reconstructed dendritic arborization possessed passive linear or active nonlinear membrane properties, we studied the effect of morphology of the dendrites on their passive electrical transfer characteristics and also on the formation of patterns of spike discharges at the output of the cell under conditions of tonic activation via uniformly distributed excitatory synapses along the dendrites. For this purpose, we calculated morphometric characteristics of the size, complexity, metric asymmetry, and function of effectiveness of somatopetal transmission of the current (with estimation of the sensitivity of this efficacy to changes in the uniform membrane conductance) for the reconstructed dendritic arborization in general and also for its apical and basal subtrees. Spatial maps of the membrane potential and intracellular calcium concentration, which corresponded to certain temporal patterns of spike discharges generated by the neuron upon different intensities of synaptic activation, were superimposed on the 3D image and dendrograms of the neuron. These maps were considered “spatial autographs” of the above patterns. The main discharge pattern included periodic two-spike bursts (dublets) generated with relatively stable intraburst interspike intervals and interburst intervals decreasing with a rise in the intensity of activation. Under conditions of intense activation, the interburst intervals became close to the intraburst intervals, so the cell began to generate continuous trains of action potentials. Such a repertoire (consisting of two patterns of the activity, periodical dublets and continuous discharges) is considerably scantier than that described earlier in pyramidal neurons of the neocortical layer 5. Under analogous conditions of activation, we observed in the latter cells a variety of patterns of output discharges of different complexities, including stochastic ones. A relatively short length of the apical dendrite subtree of layer 2/3 neurons and, correspondingly, a smaller metric asymmetry (differences between the lengths of the apical and basal dendritic branches and paths), as compared with those in layer 5 pyramidal neurons, are morphological factors responsible for the predominance of periodic spike dublets. As a result, there were two combinations of different electrical states of the sites of dendritic arborization (“spatial autographs”). In the case of dublets, these were high depolarization of the apical dendrites vs. low depolarization of the basal dendrites and a reverse combination; only the latter (reverse) combination corresponded to the case of continuous discharges. The relative simplicity and uniformity of spike patterns in the cells, apparently, promotes the predominance of network interaction in the processes of formation of the activity of pyramidal neurons of layers 2/3 and, thereby, a higher efficiency of the processes of intracortical association.     

A model for neuron firing with exponential decay of potential resulting in diffusion equations for probability density

A neuron is assumed to receive synaptic input of both excitatory and inhibitory natures from a large number of neighboring neurons; it is also assumed that a large number of such impulses are required to raise the neuron’s transmembrane potential to its threshold potential, at which it “fires” or “spikes”. The model is similar to one of Gerstein and Mandelbrot, except that in the absence of input an exponential decay of potential toward a resting level is introduced. Computational methods of determining the spike timeinterval distribution are discussed, along with the inverse problem of estimating the parameters of the system from observed spike time-interval data.     

Bifurcations of lurching waves in a thalamic neuronal network

We consider a two-layer, one-dimensional lattice of neurons; one layer consists of excitatory thalamocortical neurons, while the other is comprised of inhibitory reticular thalamic neurons. Such networks are known to support “lurching” waves, for which propagation does not appear smooth, but rather progresses in a saltatory fashion; these waves can be characterized by different spatial widths (different numbers of neurons active at the same time). We show that these lurching waves are fixed points of appropriately defined Poincaré maps, and follow these fixed points as parameters are varied. In this way, we are able to explain observed transitions in behavior, and, in particular, to show how branches with different spatial widths are linked with each other. Our computer-assisted analysis is quite general and could be applied to other spatially extended systems which exhibit this non-trivial form of wave propagation.     

Neuronal control of behavioral plasticity: the prepacemaker nucleus of weakly electric gymnotiform fish

 Gymnotiform fish of the genera Apteronotus and Eigenmannia provide an excellent vertebrate model system to study neural mechanisms controlling behavioral plasticity. These teleosts generate, by means of an electric organ, quasi-sinusoidal discharges of extremely stable frequency and waveform. Modulations consisting of transient rises in discharge frequency are produced during social encounters, and play an important role in communication. These so-called “chirps” exhibit a remarkable sexual dimorphism, as well as an enormous seasonal and individual variability. Chirping behavior is controlled by a subset of neurons in the complex of the central posterior/prepacemaker nucleus in the diencephalon. It is hypothesized that the plasticity in the performance of chirping behavior is, at least in part, governed by two mechanisms: first, by seasonally induced structural changes in dendritic morphology of neurons of the prepacemaker nucleus, thus leading to pronounced alterations in excitatory input. Second, by androgen-controlled changes in the innervation pattern of the prepacemaker nucleus by fibers expressing the neuropeptide substance P. In addition to these two dynamic processes, cells are generated continuously and at high number in the central posterior/prepacemaker nucleus during adulthood. This phenomenon may provide the basis for a “refreshment”, thus facilitating possible changes in the underlying neural network. Accepted: 17 September 1990     

Object segmentation model: analytical results and biological implications

 A simple, biologically motivated neural network for segmentation of a moving object from a visual scene is presented. The model consists of two parts: an object selection model which employs a scaling approach for receptive field sizes, and a subsequent network implementing a spotlight by means of multiplicative synapses. The network selects one object out of several, segments the rough contour of the object, and encodes the winner object's position with high accuracy. Analytical equations for the performance level of the network, e.g., for the critical distance of two objects above which they are perceived as separate, are derived. The network preferentially chooses the object with the largest angular velocity and the largest angular width. An equation for the velocity and width preferences is presented. Additionally it is shown that for certain neurons of the model, flat receptive fields are more favourable than Gaussian ones. The network exhibits performances similar to those known from amphibians. Various electrophysiological and behavioral results – e.g., the distribution of the diameters of the receptive fields of tectal neurons, of the tongue-projecting salamander Hydromantes italicus and the range of optimal prey velocities for prey catching – can be understood on the basis of the model. Received: 7 December 2000 / Accepted: 13 February 2001     

Neuronal Expression of Splice Variants of “Glial” Glutamate Transporters in Brains Afflicted by Alzheimer’s Disease: Unmasking an Intrinsic Neuronal Property

Anomalies in glutamate homeostasis may contribute to the pathological processes involved in Alzheimer’s disease (AD). Glutamate released from neurons or glial cells is normally rapidly cleared by glutamate transporters, most of which are expressed at the protein level by glial cells. However, in some patho-physiological situations, expression of glutamate transporters that are normally considered to be glial types, appears to be evoked in populations of distressed neurons. This study analysed the expression of exon-skipping forms of the three predominant excitatory amino acid (glutamate) transporters (EAATs1-3) in brains afflicted with AD. We demonstrate by immunocytochemistry in temporal cortex, the expression of these proteins particularly in limited subsets of neurons, some of which appeared to be dys-morphic. Whilst the neuronal expression of the “glial” glutamate transporters EAAT1 and EAAT2 is frequently considered to represent the abnormal and ectopic expression of such transporters, we suggest this may be a misinterpretation, since neurons such as cortical pyramidal cells normally express abundant mRNA for these EAATs (but little if any EAAT protein expression). We hypothesize instead that distressed neurons in the AD brain can turn on the translation of pre-existent mRNA pools, or suppress the degradation of alternately spliced glutamate transporter protein, leading to the “unmasking” of, rather than evoked expression of “glial” glutamate transporters in stressed neurons. Special issue article in honor of Dr. Graham Johnston.     

A new model for simple neural nets and its application in the design of a neural oscillator

The results of a previous theoretical study of a class of systems are applied for the design of neural nets which try to simulate biological behavior. Besides the models for single aperiodic and periodic neurons, a “neural oscillator” is developed which consists of two cross-excited neurons. Its response is similar to the firing pattern of certain biological neural oscillators, like the flying system of the locust. Also, by proper change of its parameters, it can be made highly irregular, providing a deterministic model for the spontaneous neural activity.     

Age changes of the carpal bones in macaques: Application of the fourier analysis on the radiographs

A method based on the Fourier analysis is proposed, which describes and analyzes the contour morphology of carpal bones by separating morphology into factors of shape and size. Here, “size” refers to the average diameter of the contour. The “shape” is expressed byshape factors which are derived from the Fourier series and the “shape” of a monkey is expressed by ashape index which is calculated fromshape factors. The age change in the morphology of the lunate and capitate ofMacaca fuscata fuscata was analyzed by this method. The development of “shape” approximately completes by 3 years of age, whereas increase in “size” begins its spurt at that age as do body weight and anterior trunk length. By applying this method to other macaque species, it was found thatM. mulatta, M. f. yakui andM. cyclopis exhibit similar patterns of growth and development of carpal bones to those ofM. f. fuscata. Patterns found inM. fascicularis differ in that its bones develop faster than in the other macaques with respect to the “shape,” but remain small with respect to the “size.”