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.     

Entrainment ranges of forced phase oscillators

In the vertebrate spinal cord, a neural circuit called the central pattern generator produces the basic locomotory rhythm. Short and long distance intersegmental connections serve to maintain coordination along the length of the body. As a way of examining the influence of such connections, we consider a model of a chain of coupled phase oscillators in which one oscillator receives a periodic forcing stimulus. For a certain range of forcing frequencies, the chain will match the stimulus frequency, a phenomenon called entrainment. Motivated by recent experiments in lampreys, we derive analytical expressions for the range of forcing frequencies that entrain the chain, and how that range depends on the forcing location. For short intersegmental connections, in which an oscillator is connected only to its nearest neighbors, we describe two ways in which entrainment is lost: internally, in which oscillators within the chain no longer oscillate at the same frequency; and externally, in which the the chain no longer has the same frequency as the forcing. By analyzing chains in which every oscillator is connected to every other oscillator (i.e., all-to-all connections), we show that the presence of connections with lengths greater than one do not necessarily change the entrainment ranges based on the nearest–neighbor model. We derive a criterion for the ratio of connection strengths under which the connections of length greater than one do not change the entrainment ranges produced in the nearest–neighbor model, provided entrainment is lost externally. However, when this criterion holds, the range of entrained frequencies is a monotonic function of forcing location, unlike experimental results, in which entrainment ranges are larger near the middle of the chain than at the ends. Numerically, we show that similar non-monotonic entrainment ranges are possible if the ratio criterion does not hold, suggesting that in the lamprey central pattern generator, intersegmental connection strengths are not a simple function of the connection length.     

Parkinsonian tremor and simplification in network dynamics

We explore the behavior of richly connected inhibitory neural networks under parameter changes that correspond to weakening of synaptic efficacies between network units, and show that transitions from irregular to periodic dynamics are common in such systems. The weakening of these connections leads to a reduction in the number of units that effectively drive the dynamics and thus to simpler behavior. We hypothesize that the multiple interconnecting loops of the brain’s motor circuitry, which involve many inhibitory connections, exhibit such transitions. Normal physiological tremor is irregular while other forms of tremor show more regular oscillations. Tremor in Parkinson’s disease, for example, stems from weakened synaptic efficacies of dopaminergic neurons in the nigro-striatal pathway, as in our general model. The multiplicity of structures involved in the production of symptoms in Parkinson’s disease and the reversibility of symptoms by pharmacological and surgical manipulation of connection parameters suggest that such a neural network model is appropriate. Furthermore, fixed points that can occur in the network models are suggestive of akinesia in Parkinson’s disease. This model is consistent with the view that normal physiological systems can be regulated by robust and richly connected feedback networks with complex dynamics, and that loss of complexity in the feedback structure due to disease leads to more orderly behavior.     

Identifying disease-associated biomarker network features through conditional graphical model

Biomarkers are often organized into networks, in which the strengths of network connections vary across subjects depending on subject-specific covariates (eg, genetic variants). Variation of network connections, as subject-specific feature variables, has been found to predict disease clinical outcome. In this work, we develop a two-stage method to estimate biomarker networks that account for heterogeneity among subjects and evaluate network's association with disease clinical outcome. In the first stage, we propose a conditional Gaussian graphical model with mean and precision matrix depending on covariates to obtain covariate-dependent networks with connection strengths varying across subjects while assuming homogeneous network structure. In the second stage, we evaluate clinical utility of network measures (connection strengths) estimated from the first stage. The second-stage analysis provides the relative predictive power of between-region network measures on clinical impairment in the context of regional biomarkers and existing disease risk factors. We assess the performance of proposed method by extensive simulation studies and application to a Huntington's disease (HD) study to investigate the effect of HD causal gene on the rate of change in motor symptom through affecting brain subcortical and cortical gray matter atrophy connections. We show that cortical network connections and subcortical volumes, but not subcortical connections are identified to be predictive of clinical motor function deterioration. We validate these findings in an independent HD study. Lastly, highly similar patterns seen in the gray matter connections and a previous white matter connectivity study suggest a shared biological mechanism for HD and support the hypothesis that white matter loss is a direct result of neuronal loss as opposed to the loss of myelin or dysmyelination.     

Survival and production in variable resource environments

A dynamic energy budget (DEB) model describes the rates at which organisms assimilate and utilize energy from food for maintenance, growth, reproduction and development. We study the dynamic behavior of one particular DEB model, Kooijman’s κ rule model, whose key assumption is that somatic and reproductive tissues are competing for energy. We assume an environment in which the food density fluctuates either periodically or stochastically (pink noise). Both types of fluctuations stimulate growth; the magnitude of the (average) increase in size depends on both the strength and duration of the fluctuations. In a stochastic environment, the risk of mortality due to starvation increases with increasing fluctuation intensity. The mean lifespan is also a function of the model parameter κ characterizing the partitioning of energy between somatic and reproductive tissues. Organisms committing a large fraction of resources to reproduction endure periods of food shortage relatively well. The effects of food fluctuations on reproduction are complex. With stochastic food, reproduction in survivors increases with increasing fluctuation intensities, but lifetime reproduction decreases. Periodic fluctuations may enhance reproduction, depending on the value of κ. Thus, a variable food supply stimulates growth, increases mortality and may enhance reproduction, depending on life history.     

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.     

Going Beyond a Mean-field Model for the Learning Cortex: Second-Order Statistics

Mean-field models of the cortex have been used successfully to interpret the origin of features on the electroencephalogram under situations such as sleep, anesthesia, and seizures. In a mean-field scheme, dynamic changes in synaptic weights can be considered through fluctuation-based Hebbian learning rules. However, because such implementations deal with population-averaged properties, they are not well suited to memory and learning applications where individual synaptic weights can be important. We demonstrate that, through an extended system of equations, the mean-field models can be developed further to look at higher-order statistics, in particular, the distribution of synaptic weights within a cortical column. This allows us to make some general conclusions on memory through a mean-field scheme. Specifically, we expect large changes in the standard deviation of the distribution of synaptic weights when fluctuation in the mean soma potentials are large, such as during the transitions between the “up” and “down” states of slow-wave sleep. Moreover, a cortex that has low structure in its neuronal connections is most likely to decrease its standard deviation in the weights of excitatory to excitatory synapses, relative to the square of the mean, whereas a cortex with strongly patterned connections is most likely to increase this measure. This suggests that fluctuations are used to condense the coding of strong (presumably useful) memories into fewer, but dynamic, neuron connections, while at the same time removing weaker (less useful) memories.     

A hemicord locomotor network of excitatory interneurons: a simulation study

Locomotor burst generation is simulated using a full-scale network model of the unilateral excitatory interneuronal population. Earlier small-scale models predicted that a population of excitatory neurons would be sufficient to produce burst activity, and this has recently been experimentally confirmed. Here we simulate the hemicord activity induced under various experimental conditions, including pharmacological activation by NMDA and AMPA as well as electrical stimulation. The model network comprises a realistic number of cells and synaptic connectivity patterns. Using similar distributions of cellular and synaptic parameters, as have been estimated experimentally, a large variation in dynamic characteristics like firing rates, burst, and cycle durations were seen in single cells. On the network level an overall rhythm was generated because the synaptic interactions cause partial synchronization within the population. This network rhythm not only emerged despite the distributed cellular parameters but relied on this variability, in particular, in reproducing variations of the activity during the cycle and showing recruitment in interneuronal populations. A slow rhythm (0.4–2 Hz) can be induced by tonic activation of NMDA-sensitive channels, which are voltage dependent and generate depolarizing plateaus. The rhythm emerges through a synchronization of bursts of the individual neurons. A fast rhythm (4–12 Hz), induced by AMPA, relies on spike synchronization within the population, and each burst is composed of single spikes produced by different neurons. The dynamic range of the fast rhythm is limited by the ability of the network to synchronize oscillations and depends on the strength of synaptic connections and the duration of the slow after hyperpolarization. The model network also produces prolonged bouts of rhythmic activity in response to brief electrical activations, as seen experimentally. The mutual excitation can sustain long-lasting activity for a realistic set of synaptic parameters. The bout duration depends on the strength of excitatory synaptic connections, the level of persistent depolarization, and the influx of Ca²⁺ ions and activation of Ca²⁺-dependent K⁺ current.     

Network Self-Organization Explains the Statistics and Dynamics of Synaptic Connection Strengths in Cortex

The information processing abilities of neural circuits arise from their synaptic connection patterns. Understanding the laws governing these connectivity patterns is essential for understanding brain function. The overall distribution of synaptic strengths of local excitatory connections in cortex and hippocampus is long-tailed, exhibiting a small number of synaptic connections of very large efficacy. At the same time, new synaptic connections are constantly being created and individual synaptic connection strengths show substantial fluctuations across time. It remains unclear through what mechanisms these properties of neural circuits arise and how they contribute to learning and memory. In this study we show that fundamental characteristics of excitatory synaptic connections in cortex and hippocampus can be explained as a consequence of self-organization in a recurrent network combining spike-timing-dependent plasticity (STDP), structural plasticity and different forms of homeostatic plasticity. In the network, associative synaptic plasticity in the form of STDP induces a rich-get-richer dynamics among synapses, while homeostatic mechanisms induce competition. Under distinctly different initial conditions, the ensuing self-organization produces long-tailed synaptic strength distributions matching experimental findings. We show that this self-organization can take place with a purely additive STDP mechanism and that multiplicative weight dynamics emerge as a consequence of network interactions. The observed patterns of fluctuation of synaptic strengths, including elimination and generation of synaptic connections and long-term persistence of strong connections, are consistent with the dynamics of dendritic spines found in rat hippocampus. Beyond this, the model predicts an approximately power-law scaling of the lifetimes of newly established synaptic connection strengths during development. Our results suggest that the combined action of multiple forms of neuronal plasticity plays an essential role in the formation and maintenance of cortical circuits.     

Corticofugal feedback can reduce the visual latency of responses to antagonistic stimuli

 A biophysically realistical model of the primary visual pathway is designed, including feedback connections from the visual cortex to the lateral geniculate nucleus (LGN) – the so-called corticofugal pathway. The model comprises up to 10 000 retina and LGN cells divided into the ON and the OFF pathway according to their contrast response characteristics. An additional 6000 cortical simple cells are modeled. Apart from the direct excitatory afferent pathway we include strong mutual inhibition between the ON and the OFF subsystems. In addition, we propose a novel type of paradoxical corticofugal connection pattern which links ON dominated cortical simple cells to OFF-center LGN cells and vice versa. In accordance with physiological findings these connections are weakly excitatory and do not interfere with the steady-state responses to constant illumination, because during the steady-state inhibition arising from the active pathway effectively silences the nonstimulated pathway. At the moment of a contrast reversal the effect of the paradoxical connection pattern comes into play and the depolarization of the previously silent channel is accelerated, leading to a latency reduction of up to 4 ms using moderate synaptic weights. With increased weights reductions of more than 10 ms can be achieved. We introduce different synaptic characteristics for the feedback (AMPA, NMDA, AMPA+NMDA) and show that the strongest latency reduction is obtained for a combination of the membrane channels (i.e., AMPA+NMDA). The effect of the proposed paradoxical connection pattern is self-regulating; because the levels of inhibition and paradoxical excitation are always driven by the same inputs (strong inhibition is counterbalanced by a stronger paradoxical excitation and vice versa). In addition, the latency reduction for a contrast inversion which ends at a small absolute contrast level (small contrast step) is stronger than the reduction for an inversion with large final contrast (large contrast step). This leads to a more pronounced reduction in the reaction times for weak stimuli. Thus, reaction time differences for different contrast steps are smoothed out. Received: 22 January 1996/Accepted in revised form: 20 May 1996     

Stability and structural constraints of random brain networks with excitatory and inhibitory neural populations

The stability of brain networks with randomly connected excitatory and inhibitory neural populations is investigated using a simplified physiological model of brain electrical activity. Neural populations are randomly assigned to be excitatory or inhibitory and the stability of a brain network is determined by the spectrum of the network’s matrix of connection strengths. The probability that a network is stable is determined from its spectral density which is numerically determined and is approximated by a spectral distribution recently derived by Rajan and Abbott. The probability that a brain network is stable is maximum when the total connection strength into a population is approximately zero and is shown to depend on the arrangement of the excitatory and inhibitory connections and the parameters of the network. The maximum excitatory and inhibitory input into a structure allowed by stability occurs when the net input equals zero and, in contrast to networks with randomly distributed excitatory and inhibitory connections, substantially increases as the number of connections increases. Networks with the largest excitatory and inhibitory input allowed by stability have multiple marginally stable modes, are highly responsive and adaptable to external stimuli, have the same total input into each structure with minimal variance in the excitatory and inhibitory connection strengths, and have a wide range of flexible, adaptable, and complex behavior.     

Columnar Architecture Improves Noise Robustness in a Model Cortical Network

Cortical columnar architecture was discovered decades ago yet there is no agreed upon explanation for its function. Indeed, some have suggested that it has no function, it is simply an epiphenomenon of developmental processes. To investigate this problem we have constructed a computer model of one square millimeter of layer 2/3 of the primary visual cortex (V1) of the cat. Model cells are connected according to data from recent paired cell studies, in particular the connection probability between pyramidal cells is inversely proportional both to the distance separating the cells and to the distance between the preferred parameters (features) of the cells. We find that these constraints, together with a columnar architecture, produce more tightly clustered populations of cells when compared to the random architecture seen in, for example, rodents. This causes the columnar network to converge more quickly and accurately on the pattern representing a particular stimulus in the presence of noise, suggesting that columnar connectivity functions to improve pattern recognition in cortical circuits. The model also suggests that synaptic failure, a phenomenon exhibited by weak synapses, may conserve metabolic resources by reducing transmitter release at these connections that do not contribute to network function.     

Bayesian inference in populations of cortical neurons: a model of motion integration and segmentation in area MT

A major issue in cortical physiology and computational neuroscience is understanding the interaction between extrinsic signals from feedforward connections and intracortical signals from lateral connections. We propose here a computational model for motion perception based on the assumption that the local cortical circuits in the medio-temporal area (area MT) implement a Bayesian inference principle. This approach establishes a functional balance between feedforward and lateral, excitatory and inhibitory, inputs. The model reproduces most of the known properties of the neurons in area MT in response to moving stimuli. It accounts for important motion perception phenomena including motion transparency, spatial and temporal integration/segmentation. While integrating several properties of previously proposed models, it makes specific testable predictions concerning, in particular, temporal properties of neurons and the architecture of lateral connections in area MT. In addition, the proposed mechanism is consistent with the known properties of local cortical circuits in area V1. This suggests that Bayesian inference may be a general feature of information processing in cortical neuron populations. Received: 3 December 1997 / Accepted in revised form: 21 July 1998     

Bistable MAP kinase activity: a plausible mechanism contributing to maintenance of late long-term potentiation

Bistability of MAP kinase (MAPK) activity has been suggested to contribute to several cellular processes, including differentiation and long-term synaptic potentiation. A recent model (Markevich NI, Hoek JB, Kholodenko BN. J Cell Biol 164: 353–359, 2004) predicts bistability due to interactions of the kinases and phosphatases in the MAPK pathway, without feedback from MAPK to earlier reactions. Using this model and enzyme concentrations appropriate for neurons, we simulated bistable MAPK activity, but bistability was present only within a relatively narrow range of activity of Raf, the first pathway kinase. Stochastic fluctuations in molecule numbers eliminated bistability for small molecule numbers, such as are expected in the volume of a dendritic spine. However, positive-feedback loops have been posited from MAPK up to Raf activation. One proposed loop in which MAPK directly activates Raf was incorporated into the model. We found that such feedback greatly enhanced the robustness of both stable states of MAPK activity to stochastic fluctuations and to parameter variations. Bistability was robust for molecule numbers plausible for a dendritic spine volume. The upper state of MAPK activity was resistant to inhibition of MEK activation for >1 h, which suggests that inhibitor experiments have not sufficed to rule out a role for persistent MAPK activity in the maintenance of long-term potentiation (LTP). These simulations suggest that persistent MAPK activity and consequent upregulation of translation may contribute to LTP maintenance and to long-term memory. Experiments using a fluorescent MAPK substrate may further test this hypothesis. feedback; bistability; memory; model; stochastic     

Self-assembly of neural networks viewed as swarm intelligence

While self-assembly is a fairly active area of research in swarm intelligence, relatively little attention has been paid to the issues surrounding the construction of network structures. In this paper we extend methods developed previously for controlling collective movements of agent teams to serve as the basis for self-assembly or “growth” of networks, using neural networks as a concrete application to evaluate our approach. Our central innovation is having network connections arise as persistent “trails” left behind moving agents, trails that are reminiscent of pheromone deposits made by agents in ant colony optimization models. The resulting network connections are thus essentially a record of agent movements. We demonstrate our model’s effectiveness by using it to produce two large networks that support subsequent learning of topographic and feature maps. Improvements produced by the incorporation of collective movements are also examined through computational experiments. These results indicate that methods for directing collective movements can be adopted to facilitate network self-assembly.     

Cooperative lattice dynamics and anomalous fluctuations of microtubules

Microtubules have been in the focus of biophysical research for several decades. However, the confusing and mutually contradictory results regarding their elasticity and fluctuations have cast doubt on their present understanding. In this paper, we present the empirical evidence for the existence of discrete guanosine diphosphate (GDP)–tubulin fluctuations between a curved and a straight configuration at room temperature as well as for conformational tubulin cooperativity. Guided by a number of experimental findings, we build the case for a novel microtubule model, with the principal result that microtubules can spontaneously form micron-sized cooperative helical states with unique elastic and dynamic features. The polymorphic dynamics of the microtubule lattice resulting from the tubulin bistability quantitatively explains several experimental puzzles, including anomalous scaling of dynamic fluctuations of grafted microtubules, their apparent length–stiffness relation, and their remarkable curved–helical appearance in general. We point out that the multistability and cooperative switching of tubulin dimers could participate in important cellular processes, and could in particular lead to efficient mechanochemical signaling along single microtubules.     

PARASITIC CASTRATION PROMOTES COEVOLUTIONARY CYCLING BUT ALSO IMPOSES A COST ON SEX

Antagonistic coevolution between hosts and parasites is thought to drive a range of biological phenomena including the maintenance of sexual reproduction. Of particular interest are conditions that produce persistent fluctuations in the frequencies of genes governing host–parasite specificity (coevolutionary cycling), as sex may be more beneficial than asexual reproduction in a constantly changing environment. Although many studies have shown that coevolutionary cycling can lead to the maintenance of sex, the effects of ecological feedbacks on the persistence of these fluctuations in gene frequencies are not well understood. Here, we use a simple deterministic model that incorporates ecological feedbacks to explore how parasitic reductions in host fecundity affect the maintenance of coevolutionary cycling. We demonstrate that parasitic castration is inherently destabilizing and may be necessary for coevolutionary cycling to persist indefinitely, but also reduces the likelihood that sexually reproducing individuals will find a fertile partner, which may select against sex. These findings suggest that castrators can play an important role in shaping host evolution and are likely to be good targets for observing fluctuations in gene frequencies that govern specificity in host–parasite interactions.     

Spiking neural network model for memorizing sequences with forward and backward recall

We present an oscillatory network of conductance based spiking neurons of Hodgkin–Huxley type as a model of memory storage and retrieval of sequences of events (or objects). The model is inspired by psychological and neurobiological evidence on sequential memories. The building block of the model is an oscillatory module which contains excitatory and inhibitory neurons with all-to-all connections. The connection architecture comprises two layers. A lower layer represents consecutive events during their storage and recall. This layer is composed of oscillatory modules. Plastic excitatory connections between the modules are implemented using an STDP type learning rule for sequential storage. Excitatory neurons in the upper layer project star-like modifiable connections toward the excitatory lower layer neurons. These neurons in the upper layer are used to tag sequences of events represented in the lower layer. Computer simulations demonstrate good performance of the model including difficult cases when different sequences contain overlapping events. We show that the model with STDP type or anti-STDP type learning rules can be applied for the simulation of forward and backward replay of neural spikes respectively.     

A neural network model of the inferior colliculus with modifiable lateral inhibitory synapses for human echolocation

 We propose a neural network model of the inferior colliculus (IC) for human echolocation. Neuronal mechanisms for human echolocation were investigated by simulating the model. The model consists of the neural networks of the central nucleus (ICc) and external nucleus (ICx) of the inferior colliculus. The neurons of the ICc receive interaural sound stimuli via multiple contralateral delay lines and a single ipsilateral delay line. The neurons of the ICc send output signals to the neurons of the ICx in a convergent manner. We stimulated the ICc with pairs of a direct sound (a sonar sound) and an echo sound (the reflection from an object). Information about the distance between the model and the object is expressed by the delay time of the echo sound with respect to the direct sound. The results presented here show that neurons of the ICc responsive to interaural onset time differences contribute to the creation of an auditory distance map in the ICx. We trained the model with various pairs of direct-echo sounds and modified synaptic connection strengths of the networks according to the Hebbian rule. It is shown that self-organized long-term depression of lateral inhibitory synaptic connections plays an important role in enhancing echolocation skills. Received: 26 November 2000 / Accepted in revised form: 16 October 2001     

Identification of candidate small-molecule therapeutics to cancer by gene-signature perturbation in connectivity mapping

Connectivity mapping is a recently developed technique for discovering the underlying connections between different biological states based on gene-expression similarities. The sscMap method has been shown to provide enhanced sensitivity in mapping meaningful connections leading to testable biological hypotheses and in identifying drug candidates with particular pharmacological and/or toxicological properties. Challenges remain, however, as to how to prioritise the large number of discovered connections in an unbiased manner such that the success rate of any following-up investigation can be maximised. We introduce a new concept, gene-signature perturbation, which aims to test whether an identified connection is stable enough against systematic minor changes (perturbation) to the gene-signature. We applied the perturbation method to three independent datasets obtained from the GEO database: acute myeloid leukemia (AML), cervical cancer, and breast cancer treated with letrozole. We demonstrate that the perturbation approach helps to identify meaningful biological connections which suggest the most relevant candidate drugs. In the case of AML, we found that the prevalent compounds were retinoic acids and PPAR activators. For cervical cancer, our results suggested that potential drugs are likely to involve the EGFR pathway; and with the breast cancer dataset, we identified candidates that are involved in prostaglandin inhibition. Thus the gene-signature perturbation approach added real values to the whole connectivity mapping process, allowing for increased specificity in the identification of possible therapeutic candidates.