Storage of correlated patterns in standard and bistable Purkinje cell models

The cerebellum has long been considered to undergo supervised learning, with climbing fibers acting as a 'teaching' or 'error' signal. Purkinje cells (PCs), the sole output of the cerebellar cortex, have been considered as analogs of perceptrons storing input/output associations. In support of this hypothesis, a recent study found that the distribution of synaptic weights of a perceptron at maximal capacity is in striking agreement with experimental data in adult rats. However, the calculation was performed using random uncorrelated inputs and outputs. This is a clearly unrealistic assumption since sensory inputs and motor outputs carry a substantial degree of temporal correlations. In this paper, we consider a binary output neuron with a large number of inputs, which is required to store associations between temporally correlated sequences of binary inputs and outputs, modelled as Markov chains. Storage capacity is found to increase with both input and output correlations, and diverges in the limit where both go to unity. We also investigate the capacity of a bistable output unit, since PCs have been shown to be bistable in some experimental conditions. Bistability is shown to enhance storage capacity whenever the output correlation is stronger than the input correlation. Distribution of synaptic weights at maximal capacity is shown to be independent on correlations, and is also unaffected by the presence of bistability.     

Multistable Decision Switches for Flexible Control of Epigenetic Differentiation

It is now recognized that molecular circuits with positive feedback can induce two different gene expression states (bistability) under the very same cellular conditions. Whether, and how, cells make use of the coexistence of a larger number of stable states (multistability) is however largely unknown. Here, we first examine how autoregulation, a common attribute of genetic master regulators, facilitates multistability in two-component circuits. A systematic exploration of these modules'' parameter space reveals two classes of molecular switches, involving transitions in bistable (progression switches) or multistable (decision switches) regimes. We demonstrate the potential of decision switches for multifaceted stimulus processing, including strength, duration, and flexible discrimination. These tasks enhance response specificity, help to store short-term memories of recent signaling events, stabilize transient gene expression, and enable stochastic fate commitment. The relevance of these circuits is further supported by biological data, because we find them in numerous developmental scenarios. Indeed, many of the presented information-processing features of decision switches could ultimately demonstrate a more flexible control of epigenetic differentiation.     

Intrinsic and network rhythmogenesis in a reduced traub model for CA3 neurons

We have developed a two-compartment, eight-variable model of a CA3 pyramidal cell as a reduction of a complex 19-compartment cable model [Traub et al, 1991]. Our reduced model segregates the fast currents for sodium spiking into a proximal, soma-like, compartment and the slower calcium and calcium-mediated currents into a dendrite-like compartment. In each model periodic bursting gives way to repetitive soma spiking as somatic injected current increases. Steady dendritic stimulation can produce periodic bursting of significantly higher frequency (8–20 Hz) than can steady somatic input (<8 Hz). Bursting in our model occurs only for an intermediate range of electronic coupling conductance. It depends on the segregation of channel types and on the coupling current that flows back-and-forth between compartments. When the soma and dendrite are tightly coupled electrically, our model reduces to a single compartment and does not burst. Network simulations with our model using excitatory AMPA and NMDA synapses (without inhibition) give results similar to those obtained with the complex cable model [Traub et al, 1991; Traub et al, 1992]. Brief stimulation of a single cell in a resting network produces multiple synchronized population bursts, with fast AMPA synapses providing the dominant synchronizing mechanism. The number of bursts increases with the level of maximal NMDA conductance. For high enough maximal NMDA conductance synchronized bursting repeats indefinitely. We find that two factors can cause the cells to desynchronize when AMPA synapses are blocked: heterogeneity of properties amongst cells and intrinsically chaotic burst dynamics. But even when cells are identical, they may synchronize only approximately rather than exactly. Since our model has a limited number of parameters and variables, we have studied its cellular and network dynamics computationally with relative ease and over wide parameter ranges. Thereby, we identify some qualitative features that parallel or are distinguished from those of other neuronal systems; e.g., we discuss how bursting here differs from that in some classical models.     

Reconstructing pathways in large genetic networks from genetic perturbations.

I present an algorithm that determines the longest path between every gene pair in an arbitrarily large genetic network from large scale gene perturbation data. The algorithm's computational complexity is O(nk(2)), where n is the number of genes in the network and k is the average number of genes affected by a genetic perturbation. The algorithm is able to distinguish a large fraction of direct regulatory interactions from indirect interactions, even if the accuracy of its input data is substantially compromised.     

Bistable responses in bacterial genetic networks: designs and dynamical consequences

A key property of living cells is their ability to react to stimuli with specific biochemical responses. These responses can be understood through the dynamics of underlying biochemical and genetic networks. Evolutionary design principles have been well studied in networks that display graded responses, with a continuous relationship between input signal and system output. Alternatively, biochemical networks can exhibit bistable responses so that over a range of signals the network possesses two stable steady states. In this review, we discuss several conceptual examples illustrating network designs that can result in a bistable response of the biochemical network. Next, we examine manifestations of these designs in bacterial master-regulatory genetic circuits. In particular, we discuss mechanisms and dynamic consequences of bistability in three circuits: two-component systems, sigma-factor networks, and a multistep phosphorelay. Analyzing these examples allows us to expand our knowledge of evolutionary design principles networks with bistable responses.     

Small Modifications to Network Topology Can Induce Stochastic Bistable Spiking Dynamics in a Balanced Cortical Model

Directed random graph models frequently are used successfully in modeling the population dynamics of networks of cortical neurons connected by chemical synapses. Experimental results consistently reveal that neuronal network topology is complex, however, in the sense that it differs statistically from a random network, and differs for classes of neurons that are physiologically different. This suggests that complex network models whose subnetworks have distinct topological structure may be a useful, and more biologically realistic, alternative to random networks. Here we demonstrate that the balanced excitation and inhibition frequently observed in small cortical regions can transiently disappear in otherwise standard neuronal-scale models of fluctuation-driven dynamics, solely because the random network topology was replaced by a complex clustered one, whilst not changing the in-degree of any neurons. In this network, a small subset of cells whose inhibition comes only from outside their local cluster are the cause of bistable population dynamics, where different clusters of these cells irregularly switch back and forth from a sparsely firing state to a highly active state. Transitions to the highly active state occur when a cluster of these cells spikes sufficiently often to cause strong unbalanced positive feedback to each other. Transitions back to the sparsely firing state rely on occasional large fluctuations in the amount of non-local inhibition received. Neurons in the model are homogeneous in their intrinsic dynamics and in-degrees, but differ in the abundance of various directed feedback motifs in which they participate. Our findings suggest that (i) models and simulations should take into account complex structure that varies for neuron and synapse classes; (ii) differences in the dynamics of neurons with similar intrinsic properties may be caused by their membership in distinctive local networks; (iii) it is important to identify neurons that share physiological properties and location, but differ in their connectivity.     

Alternative attractors in an ecological-genetic model of populations with non-overlapping generations

This study classifies and analyzes various bifurcations of fixed points of the simple model of population dynamics with its number described by Ricker’s model and intrapopulation parameters determined by a single di-allelic locus. The model considered shows such nonlinear phenomena as multistability and coexistence of alternative attractors, which can violate the simple combination of the action of evolutionary-genetic and density-dependent ecological-dynamic processes, where gene elimination is determined by the genotypes’ resource parameters and the population number stability depends on their Malthusian parameters. The most interesting pattern in this regard is existence of polymorphic attractors, when the resource parameter of heterozygotes is not maximal. It presents a clear violation of the principle of natural selection optimality, which is caused precisely by the multistable phenomena of nonlinear dynamics. By which of the alternative attractors the dynamics is characterized by depends sensitively on the initial conditions, even small external influences can became significant, as they can shift the system from one attraction basin to another, and thus fundamentally change the dynamic mode and nature of the evolutionary process.     

Deciphering the Dynamical Origin of Mixed Population during Neural Stem Cell Development

Neural stem cells (NSCs) often give rise to a mixed population of cells during differentiation. However, the dynamical origin of these mixed states is poorly understood. In this article, our mathematical modeling study demonstrates that the bone morphogenetic protein 2 (BMP2) mediated disparate differentiation dynamics of NSCs in central and peripheral nervous systems essentially function through two distinct bistable switches that are mutually interconnected via a mushroom-like bifurcation. Stochastic simulations of the model reveal that the mixed population originates due to the existence of these bistable switching regulations and that the maintenance of such mixed states depends on the level of stochastic fluctuations of the system. It further demonstrates that due to extrinsic variability, cells in an NSC population can dynamically transit from mushroom to a unique isola kind of bifurcation state, which essentially extends the range of the BMP2-driven mixed population state during differentiation. Importantly, the model predicts that by individually altering the expression level of key regulatory proteins, the NSCs can be converted entirely to a preferred phenotype for BMP2 doses that previously resulted in a mixed population. Our findings show that efficient neuronal regeneration can be achieved by systematically maneuvering the differentiation dynamics.     

Associative recognition and storage in a model network of physiological neurons

We consider a neural network model in which the single neurons are chosen to closely resemble known physiological properties. The neurons are assumed to be linked by synapses which change their strength according to Hebbian rules on a short time scale (100ms). The dynamics of the network — the time evolution of the cell potentials and the synapses — is investigated by computer simulation. As in more abstract network models (Cooper 1973; Hopfield 1982; Kohonen 1984) it is found that the local dynamics of the cell potentials and the synaptic strengths result in global cooperative properties of the network and enable the network to process an incoming flux of information and to learn and store patterns associatively. A trained net can associate missing details of a pattern, can correct wrong details and can suppress noise in a pattern. The network can further abstract the prototype from a series of patterns with variations. A suitable coupling constant connecting the dynamics of the cell potentials with the synaptic strengths is derived by a mean field approximation. This coupling constant controls the neural sensitivity and thereby avoids both extremes of the network state, the state of permanent inactivity and the state of epileptic hyperactivity.     

Gap junctions

Electrical coupling through gap junctions constitutes a mode of signal transmission between neurons (electrical synaptic transmission). Originally discovered in invertebrates and in lower vertebrates, electrical synapses have recently been reported in immature and adult mammalian nervous systems. This has renewed the interest in understanding the role of electrical synapses in neural circuit function and signal processing. The present review focuses on the role of gap junctions in shaping the dynamics of neural networks by forming electrical synapses between neurons. Electrical synapses have been shown to be important elements in coincidence detection mechanisms and they can produce complex input-output functions when arranged in combination with chemical synapses. We postulate that these synapses may also be important in redefining neuronal compartments, associating anatomically distinct cellular structures into functional units. The original view of electrical synapses as static connecting elements in neural circuits has been revised and a considerable amount of evidence suggests that electrical synapses substantially affect the dynamics of neural circuits.     

Robustness in phenotypic plasticity and heterogeneity patterns enabled by EMT networks

Epithelial-mesenchymal plasticity (EMP) is a key arm of cancer metastasis and is observed across many contexts. Cells undergoing EMP can reversibly switch between three classes of phenotypes: epithelial (E), mesenchymal (M), and hybrid E/M. While a large number of multistable regulatory networks have been identified to be driving EMP in various contexts, the exact mechanisms and design principles that enable robustness in driving EMP across contexts are not yet fully understood. Here, we investigated dynamic and structural robustness in EMP networks with regard to phenotypic heterogeneity and plasticity. We use two different approaches to simulate these networks: a computationally inexpensive, parameter-independent continuous state space Boolean model, and an ODE-based parameter-agnostic framework (RACIPE), both of which yielded similar phenotypic distributions. While the latter approach is useful for measurements of plasticity, the former model enabled us to extensively investigate robustness in phenotypic heterogeneity. Using perturbations to network topology and by varying network parameters, we show that multistable EMP networks are structurally and dynamically more robust compared with their randomized counterparts, thereby highlighting their topological hallmarks. These features of robustness are governed by a balance of positive and negative feedback loops embedded in these networks. Using a combination of the number of negative and positive feedback loops weighted by their lengths, we identified a metric that can explain the structural and dynamical robustness of these networks. This metric enabled us to compare networks across multiple sizes, and the network principles thus obtained can be used to identify fragilities in large networks without simulating their dynamics. Our analysis highlights a network topology-based approach to quantify robustness in the phenotypic heterogeneity and plasticity emergent from EMP networks.     

A neural network model for stroboscopic alternative motion

A neural network which models multistable perception is presented. The network consists of sensor and inner neurons. The dynamics is established by a stochastic neuronal dynamics, a formal Hebb-type coupling dynamics and a resource mechanism that corresponds to saturation effects in perception. From this a system of coupled differential equations is derived and analyzed. Single stimuli are bound to exactly one percept, even in ambiguous situations where multistability occurs. The network exhibits discontinuous as well as continuous phase transitions and models various empirical findings, including the percepts of succession, alternative motion and simultaneity; the percept of oscillation is explained by oscillating percepts at a continuous phase transition. Received: 13 September 1995 / Accepted: 3 June 1996     

An Appropriate Bounded Invariant Region for a Bistable Reaction-Diffusion Model of the Caspase-3/8 Feedback Loop

The apoptotic caspase-3/8 feedback loop describes the core of the extrinsic pro-apoptotic signaling pathway, an essential part of apoptosis. Latter is a prototype of the programmed cell death, which enables organisms to remove damaged or infected cells. The reaction network of the caspase-3/8 feedback loop in a single cell is modeled by a reaction-diffusion system, which shows a bistable behavior. In this work, we present an appropriate bounded invariant region for the bistable reaction-diffusion system in order to theoretically confirm that diffusion rapidly balances the concentrations of the different caspase types. This justifies the decomposition of the dynamics into a diffusion dominated part on a very short time scale and a pure reaction driven dynamics on a large time scale.     

Modulation of Oscillatory Synchronization in an Interneuronal Network under the Influence of Tonic GABA-ergic Inhibition: a Model Study

The influence of a tonic GABA-ergic current on the processes of network synchronization was examined using a computer model of the neural network with shunting GABA-ergic synapses and tonic excitation that initiated spiking. The tonic inhibitory current was characterized by two parameters, the reversal potential and the conductance introduced. We found that tonic current with a reversal potential more negative than the threshold for spike generation reduces the network spiking frequency and synchronization. A monotonic decrease in the network synchronization with augmentation of the tonic current conductance was shown. We also found that a particular range of tonic current conductance leads to a bistable character of the network dynamics. Depending on the initial conditions of the network examined, spontaneous synchronous oscillations similar to epileptiform activity could appear.     

Two-Cell to N-Cell Heterogeneous,Inhibitory Networks: Precise Linking of Multistable and Coherent Properties

Inhibitory networks are now recognized as being the controllers of several brain rhythms. However, experimental work with inhibitory cells is technically difficult not only because of their smaller percentage of the neuronal population, but also because of their diverse properties. As such, inhibitory network models with tight links to the experimental data are needed to understand their contributions to population rhythms. However, mathematical analyses of network models with more than two cells is challenging when the cellular models involve biophysical details. We use bifurcation analyses and simulations to show that two-cell analyses can quantitatively predict N-cell (N = 20, 50, 100) network dynamics for heterogeneous, inhibitory networks. Interestingly, multistable states in the two-cell system are manifest as different and distinct coherent network patterns in the N-cell networks for the same parameter sets.     

Backward bifurcation of an epidemic model with treatment

An epidemic model with a limited resource for treatment is proposed to understand the effect of the capacity for treatment. It is assumed that treatment rate is proportional to the number of infectives below the capacity and is a constant when the number of infectives is greater than the capacity. It is found that a backward bifurcation occurs if the capacity is small. It is also found that there exist bistable endemic equilibria if the capacity is low.     

How to record a million synaptic weights in a hippocampal slice

A key step toward understanding the function of a brain circuit is to find its wiring diagram. New methods for optical stimulation and optical recording of neurons make it possible to map circuit connectivity on a very large scale. However, single synapses produce small responses that are difficult to measure on a large scale. Here I analyze how single synaptic responses may be detectable using relatively coarse readouts such as optical recording of somatic calcium. I model a network consisting of 10,000 input axons and 100 CA1 pyramidal neurons, each represented using 19 compartments with voltage-gated channels and calcium dynamics. As single synaptic inputs cannot produce a measurable somatic calcium response, I stimulate many inputs as a baseline to elicit somatic action potentials leading to a strong calcium signal. I compare statistics of responses with or without a single axonal input riding on this baseline. Through simulations I show that a single additional input shifts the distribution of the number of output action potentials. Stochastic resonance due to probabilistic synaptic release makes this shift easier to detect. With ~80 stimulus repetitions this approach can resolve up to 35% of individual activated synapses even in the presence of 20% recording noise. While the technique is applicable using conventional electrical stimulation and extracellular recording, optical methods promise much greater scaling, since the number of synapses scales as the product of the number of inputs and outputs. I extrapolate from current high-speed optical stimulation and recording methods, and show that this approach may scale up to the order of a million synapses in a single two-hour slice-recording experiment.     

On the relationship between fractal geometry of space and time in which a system of interacting cells exists and dynamics of gene expression.

We report that both space and time, in which a system of interacting cells exists, possess fractal structure. Each single cell of the system can restore the hierarchical organization and dynamic features of the entire tumor. There is a relationship between dynamics of gene expression and connectivity (i.e., interconnectedness which denotes the existence of complex, dynamic relationships in a population of cells leading to the emergence of global features in the system that would never appear in a single cell existing out of the system). Fractal structure emerges owing to non-bijectivity of dynamic cellular network of genes and their regulatory elements. It disappears during tumor progression. This latter state is characterized by damped dynamics of gene expression, loss of connectivity, loss of collectivity (i.e., capability of the interconnected cells to interact in a common mode), and metastatic phenotype. Fractal structure of both space and time is necessary for a cellular system to self-organize. Our findings indicate that results of molecular studies on gene expression should be interpreted in terms of space-time geometry of the cellular system. In particular, the dynamics of gene expression in cancer cells existing in a malignant tumor is not identical with the dynamics of gene expression in the same cells cultured in the monolayer system.