Signal propagation in feedforward neuronal networks with unreliable synapses

In this paper, we systematically investigate both the synfire propagation and firing rate propagation in feedforward neuronal network coupled in an all-to-all fashion. In contrast to most earlier work, where only reliable synaptic connections are considered, we mainly examine the effects of unreliable synapses on both types of neural activity propagation in this work. We first study networks composed of purely excitatory neurons. Our results show that both the successful transmission probability and excitatory synaptic strength largely influence the propagation of these two types of neural activities, and better tuning of these synaptic parameters makes the considered network support stable signal propagation. It is also found that noise has significant but different impacts on these two types of propagation. The additive Gaussian white noise has the tendency to reduce the precision of the synfire activity, whereas noise with appropriate intensity can enhance the performance of firing rate propagation. Further simulations indicate that the propagation dynamics of the considered neuronal network is not simply determined by the average amount of received neurotransmitter for each neuron in a time instant, but also largely influenced by the stochastic effect of neurotransmitter release. Second, we compare our results with those obtained in corresponding feedforward neuronal networks connected with reliable synapses but in a random coupling fashion. We confirm that some differences can be observed in these two different feedforward neuronal network models. Finally, we study the signal propagation in feedforward neuronal networks consisting of both excitatory and inhibitory neurons, and demonstrate that inhibition also plays an important role in signal propagation in the considered networks.     

Control of multistability in ring circuits of oscillators

The essential dynamics of some biological central pattern generators (CPGs) can be captured by a model consisting of N neurons connected in a ring. These circuits, like many oscillatory nonlinear circuits of sufficient complexity, are capable of multistability, that is, of generating different firing patterns distinguished by the phasic relationships between the firing in each circuit element (neuron). Moreover, a shift in firing pattern can be induced by a transient perturbation. A systematic approach, based on phase-response curve (PRC) theory, was used to determine the optimum timing for perturbations that induce a shift in the firing pattern. The first step was to visualize the solution space of the ring circuit, including the attractive basins for each stable firing pattern; this was possible using the relative phase of N−1 oscillators, with respect to an arbitrarily selected reference oscillator, as coordinate axes. The trajectories in this phase space were determined using an iterative mapping based only on the PRCs of the uncoupled component oscillators; this algorithm was called a circuit emulator. For an accurate mapping of the attractive basin of each pattern exhibited by the ring circuit, the emulator had to take into account the effect of a perturbation or input on the timing of two bursts following the onset of the perturbation, rather than just one. The visualization of the attractive basins for rings of two, three, and four oscillators enabled the accurate prediction of the amounts of phase resetting applied to up to N−1 oscillators within a cycle that would induce a transition from any pattern to any another pattern. Finally, the timing and synaptic characterization of an input called the switch signal was adjusted to produce the desired amount of phase resetting. Received: 29 May 1998 / Accepted in revised form: 18 September 1998     

An isomorphic mapping hypothesis of the grid representation

We introduce a grid cell microcircuit hypothesis. We propose the ‘grid in the world’ (evident in grid cell discharges) is generated by a ‘grid in the cortex’. This cortical grid is formed by patches of calbindin-positive pyramidal neurons in layer 2 of medial entorhinal cortex (MEC). Our isomorphic mapping hypothesis assumes three types of isomorphism: (i) metric correspondence of neural space (the two-dimensional cortical sheet) and the external two-dimensional space within patches; (ii) isomorphism between cellular connectivity matrix and firing field; (iii) isomorphism between single cell and population activity. Each patch is a grid cell lattice arranged in a two-dimensional map of space with a neural : external scale of approximately 1 : 2000 in the dorsal part of rat MEC. The lattice behaves like an excitable medium with neighbouring grid cells exciting each other. Spatial scale is implemented as an intrinsic scaling factor for neural propagation speed. This factor varies along the dorsoventral cortical axis. A connectivity scheme of the grid system is described. Head direction input specifies the direction of activity propagation. We extend the theory to neurons between grid patches and predict a rare discharge pattern (inverted grid cells) and the relative location and proportion of grid cells and spatial band cells.     

Self-stabilization of neuronal networks

Between the extreme views concerning ontogenesis (genetic vs. environmental determination), we use a moderate approach: a somehow pre-established neuronal model network reacts to activity deviations (reflecting input to be compensated), and stabilizes itself during a complex feed-back process. Morphogenesis is based on an algorithm formalizing the compensation theory of synaptogenesis (Wolff and Wagner 1983). This algorithm is applied to randomly connected McCulloch-Pitts networks that are able to maintain oscillations of their activity patterns over time. The algorithm can lead to networks which are morphogenetically stable but preserve self-maintained oscillations in activity. This is in contrast to most of the current models of synaptogenesis and synaptic modification based on Hebbian rules of plasticity. Hebbian networks are morphogenetically unstable without additional assumptions. The effects of compensation on structural and functional properties of the networks are described. It is concluded that the compensation theory of synaptogenesis can account for the development of morphogenetically stable neuronal networks out of randomly connected networks via selective stabilization and elimination of synapses.The logic of the compensation algorithm is based on experimental results. The present paper shows that the compensation theory can not only predict the behavior of synaptic populations (Wagner and Wolff, in preparation), but it can also describe the behavior of neurons interconnected in a network, with the resulting additional system properties. The neuronal interactions-leading to equilibrium in certain cases-are a self-organizing process in the sense that all decisions are performed on the individual cell level without knowing the overall network situation or goal.     

Modulation in firing pattern and oscillation in nerve cells of Lymnaea during network reconstruction

The modulation and reconstruction of the cardio-respiratory neural circuit of Lymnaea stagnalis L. was compared to that of Helix ponatia L. where the input variation and signal molecules were found to have primary importance in network reorganization. From the cardio-respiratory circuit only neurons connected by afferent or efferent pathways to the peripheral chemosensory organ, the osphradium, were used. It was shown that, the general principles of the network reorganization is similar in the two species. The firing pattern of the neurons altered in Lymnaea depending on the input activation or presence of signal molecules in the vicinity of the neurons. The responses of the neurons to the same sensory information, originating from osphradium varied depending on their firing patterns. On central neurones the generation of phasic pattern and/or oscillation was an indicator of network disintegration leading to insensibility to the osphradial sensory inputs. Co-application of signal molecules (5HT, DA, GABA with opioid peptides) to the neurons caused a phasic firing pattern and/or oscillation leading to disintegration of one network and activation of another one. The effect of mu-opioid peptides on GABA-induced and voltage activated ion currents were shown to be the cellular target in reconstruction of neural networks in Lymnaea. The neural network reconstruction in vertebrate brain evoked by signal molecules can be compared to that observed in the identified network of Lymnaea stagnalis making this latter a useful model in further studies, too.     

The hippocampus as a cognitive graph

A theory of cognitive mapping is developed that depends only on accepted properties of hippocampal function, namely, long-term potentiation, the place cell phenomenon, and the associative or recurrent connections made among CA3 pyramidal cells. It is proposed that the distance between the firing fields of connected pairs of CA3 place cells is encoded as synaptic resistance (reciprocal synaptic strength). The encoding occurs because pairs of cells with coincident or overlapping fields will tend to fire together in time, thereby causing a decrease in synaptic resistance via long-term potentiation; in contrast, cells with widely separated fields will tend never to fire together, causing no change or perhaps (via long-term depression) an increase in synaptic resistance. A network whose connection pattern mimics that of CA3 and whose connection weights are proportional to synaptic resistance can be formally treated as a weighted, directed graph. In such a graph, a "node" is assigned to each CA3 cell and two nodes are connected by a "directed edge" if and only if the two corresponding cells are connected by a synapse. Weighted, directed graphs can be searched for an optimal path between any pair of nodes with standard algorithms. Here, we are interested in finding the path along which the sum of the synaptic resistances from one cell to another is minimal. Since each cell is a place cell, such a path also corresponds to a path in two-dimensional space. Our basic finding is that minimizing the sum of the synaptic resistances along a path in neural space yields the shortest (optimal) path in unobstructed two-dimensional space, so long as the connectivity of the network is great enough. In addition to being able to find geodesics in unobstructed space, the same network enables solutions to the "detour" and "shortcut" problems, in which it is necessary to find an optimal path around a newly introduced barrier and to take a shorter path through a hole opened up in a preexisting barrier, respectively. We argue that the ability to solve such problems qualifies the proposed hippocampal object as a cognitive map. Graph theory thus provides a sort of existence proof demonstrating that the hippocampus contains the necessary information to function as a map, in the sense postulated by others (O'Keefe, J., and L. Nadel. 1978. The Hippocampus as a Cognitive Map. Clarendon Press, Oxford, UK). It is also possible that the cognitive mapping functions of the hippocampus are carried out by parallel graph searching algorithms implemented as neural processes. This possibility has the great attraction that the hippocampus could then operate in much the same way to find paths in general problem space; it would only be necessary for pyramidal cells to exhibit a strong nonpositional firing correlate.     

Emergence of preferred firing sequences in large spiking neural networks during simulated neuronal development

Two main processes concurrently refine the nervous system over the course of development: cell death and selective synaptic pruning. We simulated large spiking neural networks (100 x 100 neurons "at birth") characterized by an early developmental phase with cell death due to excessive firing rate, followed by the onset of spike timing dependent synaptic plasticity (STDP), driven by spatiotemporal patterns of stimulation. The cell death affected the inhibitory units more than the excitatory units during the early developmental phase. The network activity showed the appearance of recurrent spatiotemporal firing patterns along the STDP phase, thus suggesting the emergence of cell assemblies from the initially randomly connected networks. Some of these patterns were detected throughout the simulation despite the activity-driven network modifications while others disappeared.     

Elastic network modeling of cellular networks unveils sensor and effector genes that control information flow

The high-level organization of the cell is embedded in indirect relationships that connect distinct cellular processes. Existing computational approaches for detecting indirect relationships between genes typically consist of propagating abstract information through network representations of the cell. However, the selection of genes to serve as the source of propagation is inherently biased by prior knowledge. Here, we sought to derive an unbiased view of the high-level organization of the cell by identifying the genes that propagate and receive information most effectively in the cell, and the indirect relationships between these genes. To this aim, we adapted a perturbation-response scanning strategy initially developed for identifying allosteric interactions within proteins. We deployed this strategy onto an elastic network model of the yeast genetic interaction profile similarity network. This network revealed a superior propensity for information propagation relative to simulated networks with similar topology. Perturbation-response scanning identified the major distributors and receivers of information in the network, named effector and sensor genes, respectively. Effectors formed dense clusters centrally integrated into the network, whereas sensors formed loosely connected antenna-shaped clusters and contained genes with previously characterized involvement in signal transduction. We propose that indirect relationships between effector and sensor clusters represent major paths of information flow between distinct cellular processes. Genetic similarity networks for fission yeast and human displayed similarly strong propensities for information propagation and clusters of effector and sensor genes, suggesting that the global architecture enabling indirect relationships is evolutionarily conserved across species. Our results demonstrate that elastic network modeling of cellular networks constitutes a promising strategy to probe the high-level organization and cooperativity in the cell.     

Reliable short-term memory in the trion model: toward a cortical language and grammar

 It has previously been shown that Hebb learning in a single column in the trion model of cortical organization occurs by selection. Motivated by von Neumann's solution for obtaining reliability and by models of circulating cortical activity, we introduce Hebb intercolumnar couplings to achieve dramatic enhancements in reliability in the firing of connected columns. In order for these enhancements to occur, specific temporal phase differences must exist between the same inherent spatial-temporal memory patterns in connected columns. We then generalize the criteria of large enhancements in the global firing of the entire connected columnar network to investigate the case when different inherent memory patterns are in the columns. The spatial rotations as well as the temporal phases now are crucial. Only certain combinations of inherent memory patterns meet these criteria with the symmetry properties playing a major role. The columnar order of these memory patterns not in the same symmetry family can be extremely important. This yields the first higher-level architecture of a cortical language and grammar within the trion model. The implications of this result with regard to an innate human language and grammar are discussed. Received: 14 June 2000 / Accepted in revised form: 25 July 2000     

Signal propagation of the MAPK cascade in Xenopus oocytes: role of bistability and ultrasensitivity for a mixed problem

The MAPK signaling cascade is nowadays understood as a network module highly conserved across species. Its main function is to transfer a signal arriving at the plasma membrane to the cellular interior. Current understanding of ‘how’ this is achieved involves the notions of ultrasensitivity and bistability which relate to the nonlinear dynamics of the biochemical network, ignoring spatial aspects. Much less, indeed, is so far known about the propagation of the signal through the cytoplasm. In this work we formulate, starting from a Michaelis–Menten model for the MAPK cascade in Xenopus oocytes, a reaction-diffusion model of the cascade. We study this model in one space dimension. Basing ourselves on previous general results on reaction diffusion models, we particularly study for our model the conditions for signal propagation. We show that the existence of a propagating front depends sensitively on the initial and boundary conditions at the plasma membrane. Possible biological consequences of this finding are discussed.     

Probability distributions associated with distinct hits on targets

Some probability distributions connected with distinct hits on targets, using two different firing schemes, are developed. It is assumed that any shot has a probabilityp, not necessarily unity, of hitting the target at which it was aimed. The development uses a well-known expression for the probability that exactlyt ofN possible events occur simultaneously. Some of the formulae developed here include as special cases the probabilities derived separately and by more complicated arguments in papers by N. Rashevsky. (Bull. Math. Biophysics,17, 45–50, 1955) and A. Rapoport (Bull. Math. Biophysics,13, 133–38, 1951).     

Learning and evolution in bacterial taxis: an operational amplifier circuit modeling the computational dynamics of the prokaryotic 'two component system' protein network

Adaptive behavior in unicellular organisms (i.e., bacteria) depends on highly organized networks of proteins governing purposefully the myriad of molecular processes occurring within the cellular system. For instance, bacteria are able to explore the environment within which they develop by utilizing the motility of their flagellar system as well as a sophisticated biochemical navigation system that samples the environmental conditions surrounding the cell, searching for nutrients or moving away from toxic substances or dangerous physical conditions. In this paper we discuss how proteins of the intervening signal transduction network could be modeled as artificial neurons, simulating the dynamical aspects of the bacterial taxis. The model is based on the assumption that, in some important aspects, proteins can be considered as processing elements or McCulloch-Pitts artificial neurons that transfer and process information from the bacterium's membrane surface to the flagellar motor. This simulation of bacterial taxis has been carried out on a hardware realization of a McCulloch-Pitts artificial neuron using an operational amplifier. Based on the behavior of the operational amplifier we produce a model of the interaction between CheY and FliM, elements of the prokaryotic two component system controlling chemotaxis, as well as a simulation of learning and evolution processes in bacterial taxis. On the one side, our simulation results indicate that, computationally, these protein 'switches' are similar to McCulloch-Pitts artificial neurons, suggesting a bridge between evolution and learning in dynamical systems at cellular and molecular levels and the evolutive hardware approach. On the other side, important protein 'tactilizing' properties are not tapped by the model, and this suggests further complexity steps to explore in the approach to biological molecular computing.     

Delayed reverberation through time windows as a key to cerebellar function

We present a functional model of the cerebellum comprising cerebellar cortex, inferior olive, deep cerebellar nuclei, and brain stem nuclei. The discerning feature of the model being time coding, we consistently describe the system in terms of postsynaptic potentials, synchronous action potentials, and propagation delays. We show by means of detailed single-neuron modeling that (i) Golgi cells can fulfill a gating task in that they form short and well-defined time windows within which granule cells can reach firing threshold, thus organizing neuronal activity in discrete `time slices', and that (ii) rebound firing in cerebellar nuclei cells is a robust mechanism leading to a delayed reverberation of Purkinje cell activity through cerebellar-reticular projections back to the cerebellar cortex. Computer simulations of the whole cerebellar network consisting of several thousand neurons reveal that reverberation in conjunction with long-term plasticity at the parallel fiber-Purkinje cell synapses enables the system to learn, store, and recall spatio-temporal patterns of neuronal activity. Climbing fiber spikes act both as a synchronization and as a teacher signal, not as an error signal. They are due to intrinsic oscillatory properties of inferior olivary neurons and to delayed reverberation within the network. In addition to clear experimental predictions the present theory sheds new light on a number of experimental observation such as the synchronicity of climbing fiber spikes and provides a novel explanation of how the cerebellum solves timing tasks on a time scale of several hundreds of milliseconds. Received: 23 July 1999 / Accepted in revised form: 31 August 1999     

Integration of cellular signals in chattering environments

Cells are constantly exposed to fluctuating environmental conditions. External signals are sensed, processed and integrated by cellular signal transduction networks, which translate input signals into specific cellular responses by means of biochemical reactions. These networks have a complex nature, and we are still far from having a complete characterization of the process through which they integrate information, specially given the noisy environment in which that information is embedded. Guided by the many instances of constructive influences of noise that have been reported in the physical sciences in the last decades, here we explore how multiple signals are integrated in an eukaryotic cell in the presence of background noise, or chatter. To that end, we use a Boolean model of a typical human signal transduction network. Despite its complexity, we find that the network is able to display simple patterns of signal integration. Furthermore, our computational analysis shows that these integration patterns depend on the levels of fluctuating background activity carried by other cell inputs. Taken together, our results indicate that signal integration is sensitive to environmental fluctuations, and that this background noise effectively determines the information integration capabilities of the cell.     

On looking for neural networks and “cell assemblies” that underlie behavior: II. Neural realization of the mathematical model

Part I (P. H. Greene,Bull. Math. Biophysics,24, 247–275, 1962) discussed a number of formal properties of animal behavior, and presented evidence that these properties would follow naturally from a model in which patterns of neural activity in perception or motor action constituted the resonant responses of linear neural networks. Equations were derived for parameters characterizing networks which would possess desired resonant responses. These equations expressed purely mathematical requirements. The present paper shows that a simple neural model would be entirely adequate to meet these requirements. According to this model, an input locus may become functionally connected to a particular resonant response mode by firing at a frequency which comes to approach the resonant frequency of that mode. The information in a complicated “cell assembly” of the type considered could be transmitted through a nerve tract by a very simple frequency code. One neurological guess is that frequency-coded inputs excite the transients in dendritic networks. If the amplitude of the pattern becomes large, as it would near resonance, the all-or-none axonal response would become excited. This axonal response would tend to augment resonant patterns and disrupt other patterns, for a reason inherent in any linear network. Since resonant responses are automatically present in any linear network, unless special processes suppress them, they must have led to overt behavior in animals first possessing such networks. Evolution either suppressed this feature or exploited it. Since its properties resemble those of animal behavior, the latter might be suspected. Some implications are presented regarding what a physiologist might have to look for when he studies a neural system. This research was supported by the Office of Naval Research under Contract No. Nonr 2121(17) NR 049-148. Reproduction in whole or in part is permitted for any purpose of the United States Government.     

Cell division in Escherichia coli septation during synchronous growth

Summary The capacity of E. coli B/r to support recombination and complementation between T4am phages during its life cycle has been analyzed in order to get information on the mechanism of cell division. It was found that a decrease in recombinants and complementation events, which is expected at the time of cell compartmentalization coincides with physical cell separation. Therefore, we conclude that the two halves of a dividing cell remain connected until a very late stage of the division period, thus allowing exchange of DNA and protein molecules.When a synchronized culture of E. coli B/r is infected at different cell age with phage T4, the number of surviving cells increases 10 min prior to cell division. At this time the cells are separated into two independent targets for killing by the phage, although there is still free exchange of DNA and proteins within the whole cell. The localized action of murein metabolizing enzymes at the site of subsequent cell division is likely to create a barriere within the cell envelope that prevents the propagation of the phage killing signal.     

Coordinated intercellular calcium waves induced by noradrenaline in rat hepatocytes: dual control by gap junction permeability and agonist.

Calcium-mobilizing agonists induce intracellular Ca2+ concentration ([Ca2+]i) changes thought to trigger cellular responses. In connected cells, rises in [Ca2+]i can propagate from cell to cell as intercellular Ca2+ waves, the mechanisms of which are not elucidated. Using fura2-loaded rat hepatocytes, we studied the mechanisms controlling coordination and intercellular propagation of noradrenaline-induced Ca2+ signals. Gap junction blockade with 18 alpha-glycyrrhetinic acid resulted in a loss of coordination between connected cells. We found that second messengers and [Ca2+]i rises in one hepatocyte cannot trigger Ca2+ responses in connected cells, suggesting that diffusion across gap junctions, while required for coordination, is not sufficient by itself for the propagation of intercellular Ca2+ waves. In addition, our experiments revealed functional differences between noradrenaline-induced Ca2+ signals in connected hepatocytes. These results demonstrate that intercellular Ca2+ signals in multicellular systems of rat hepatocytes are propagated and highly organized through complex mechanisms involving at least three factors. First, gap junction coupling ensures coordination of [Ca2+]i oscillations between the different cells; second, the presence of hormone at each hepatocyte is required for cell-cell Ca2+ signal propagation; and third, functional differences between adjacent connected hepatocytes could allow a 'pacemaker-like' intercellular spread of Ca2+ waves.     

Co-existent activity patterns in inhibitory neuronal networks with short-term synaptic depression

A network of two neurons mutually coupled through inhibitory synapses that display short-term synaptic depression is considered. We show that synaptic depression expands the number of possible activity patterns that the network can display and allows for co-existence of different patterns. Specifically, the network supports different types of n-m anti-phase firing patterns, where one neuron fires n spikes followed by the other neuron firing m spikes. When maximal synaptic conductances are identical, n-n anti-phase firing patterns are obtained and there are conductance intervals over which different pairs of these solutions co-exist. The multitude of n-m anti-phase patterns and their co-existence are not found when the synapses are non-depressing. Geometric singular perturbation methods for dynamical systems are applied to the original eight-dimensional model system to derive a set of one-dimensional conditions for the existence and co-existence of different anti-phase solutions. The generality and validity of these conditions are demonstrated through numerical simulations utilizing the Hodgkin-Huxley and Morris-Lecar neuronal models.     

Diversity of Intrinsic Frequency Encoding Patterns in Rat Cortical Neurons—Mechanisms and Possible Functions

Extracellular recordings of single neurons in primary and secondary somatosensory cortices of monkeys in vivo have shown that their firing rate can increase, decrease, or remain constant in different cells, as the external stimulus frequency increases. We observed similar intrinsic firing patterns (increasing, decreasing or constant) in rat somatosensory cortex in vitro, when stimulated with oscillatory input using conductance injection (dynamic clamp). The underlying mechanism of this observation is not obvious, and presents a challenge for mathematical modelling. We propose a simple principle for describing this phenomenon using a leaky integrate-and-fire model with sinusoidal input, an intrinsic oscillation and Poisson noise. Additional enhancement of the gain of encoding could be achieved by local network connections amongst diverse intrinsic response patterns. Our work sheds light on the possible cellular and network mechanisms underlying these opposing neuronal responses, which serve to enhance signal detection.     

A model for repetitive firing in neurons

This paper describes a model for the generation of repetitive firing patterns in single neurons to be used as a module in large-scale network simulation studies. The model is based on the combination of extended versions of Hill's model for accomodation and of Kernell's model for adaptation. Both digital computer and electronic circuit realizations of the model are presented. The model is shown to produce strength-duration curves for accomodation which are compatible with available data from real neurons. Both “high ceiling” and “low ceiling” cell types can be matched by adjusting parameters in the model. An equation relating steady-state firing rate to amplitude of applied steady current is presented which includes the accumulation of potassium conductance changes with repetitive firing. The occurence of phasic and tonic responses to step stimulation is mapped in the parameter space of the model. Several representative response patterns to irregular inputs are presented.