Modelling spatiotemporal olfactory data in two steps: from binary to Hodgkin-Huxley neurones

Network models of synchronously updated McCulloch-Pitts neurones exhibit complex spatiotemporal patterns that are similar to activities of biological neurones in phase with a periodic local field potential, such as those observed experimentally by Wehr and Laurent (1996, Nature 384, 162-166) in the locust olfactory pathway. Modelling biological neural nets with networks of simple formal units makes the dynamics of the model analytically tractable. It is thus possible to determine the constraints that must be satisfied by its connection matrix in order to make its neurones exhibit a given sequence of activity (see, for instance, Quenet et al., 2001, Neurocomputing 38-40, 831-836). In the present paper, we address the following question: how can one construct a formal network of Hodgkin-Huxley (HH) type neurones that reproduces experimentally observed neuronal codes? A two-step strategy is suggested in the present paper: first, a simple network of binary units is designed, whose activity reproduces the binary experimental codes; second, this model is used as a guide to design a network of more realistic formal HH neurones. We show that such a strategy is indeed fruitful: it allowed us to design a model that reproduces the Wehr-Laurent olfactory codes, and to investigate the robustness of these codes to synaptic noise.     

Oscillation and coding in a formal neural network considered as a guide for plausible simulations of the insect olfactory system

For the analysis of coding mechanisms in the insect olfactory system, a fully connected network of synchronously updated McCulloch and Pitts neurons (MC-P type) was developed [Quenet, B., Horn, D., 2003. The dynamic neural filter: a binary model of spatio-temporal coding. Neural Comput. 15 (2), 309-329]. Considering the update time as an intrinsic clock, this "Dynamic Neural Filter" (DNF), which maps regions of input space into spatio-temporal sequences of neuronal activity, is able to produce exact binary codes extracted from the synchronized activities recorded at the level of projection neurons (PN) in the locust antennal lobe (AL) in response to different odors [Wehr, M., Laurent, G., 1996. Odor encoding by temporal sequences of firing in oscillating neural assemblies. Nature 384, 162-166]. Here, in a first step, we separate the populations of PN and local inhibitory neurons (LN) and use the DNF as a guide for simulations based on biological plausible neurons (Hodgkin-Huxley: H-H type). We show that a parsimonious network of 10 H-H neurons generates action potentials whose timing represents the required codes. In a second step, we construct a new type of DNF in order to study the population dynamics when different delays are taken into account. We find synaptic matrices which lead to both the emergence of robust oscillations and spatio-temporal patterns, using a formal criterion, based on a Normalized Euclidian Distance (NED), in order to measure the use of the temporal dimension as a coding dimension by the DNF. Similarly to biological PN, the activity of excitatory neurons in the model can be both phase-locked to different cycles of oscillations which remind local field potential (LFP), and nevertheless exhibit dynamic behavior complex enough to be the basis of spatio-temporal codes.     

Shape of the auto- and cross-correlation histograms of the spike trains of 2 neurons with a common monosynaptic input

Form of auto- and cross-correlation histograms of impulse trains of two neurones with a common monosynaptic input from the third one were studied by methods of modelling of neuronal interaction--biomathematical (computer controlled experiment on molluscs neurones) and mathematical--in wide physiological ranges of values of parameters characterizing properties and conditions of functioning of neurones and synapses. In conditions typical of the central nervous system of mammals (but not invertebrates), when neurones are subjected to intensive random afferent synaptic bombardment and reveal no pace-maker properties, each of the possible type of common monosynaptic inputs to two neurones--excitatory, inhibitory or inhibitory-excitatory--is manifestated in cross-correlation histogram of their impulse trains in a specific way.     

Pacemaker and network mechanisms of rhythm generation: cooperation and competition

The origin of rhythmic activity in brain circuits and CPG-like motor networks is still not fully understood. The main unsolved questions are (i) What are the respective roles of intrinsic bursting and network based dynamics in systems of coupled heterogeneous, intrinsically complex, even chaotic, neurons? (ii) What are the mechanisms underlying the coexistence of robustness and flexibility in the observed rhythmic spatio-temporal patterns? One common view is that particular bursting neurons provide the rhythmogenic component while the connections between different neurons are responsible for the regularisation and synchronisation of groups of neurons and for specific phase relationships in multi-phasic patterns. We have examined the spatio-temporal rhythmic patterns in computer-simulated motif networks of H-H neurons connected by slow inhibitory synapses with a non-symmetric pattern of coupling strengths. We demonstrate that the interplay between intrinsic and network dynamics features either cooperation or competition, depending on three basic control parameters identified in our model: the shape of intrinsic bursts, the strength of the coupling and its degree of asymmetry. The cooperation of intrinsic dynamics and network mechanisms is shown to correlate with bistability, i.e., the coexistence of two different attractors in the phase space of the system corresponding to different rhythmic spatio-temporal patterns. Conversely, if the network mechanism of rhythmogenesis dominates, monostability is observed with a typical pattern of winnerless competition between neurons. We analyse bifurcations between the two regimes and demonstrate how they provide robustness and flexibility to the network performance.     

Modelling corticothalamic feedback and the gating of the thalamus by the cerebral cortex

Morphological studies have shown that excitatory synapses from the cortex constitute the major source of synapses in the thalamus. However, the effect of these corticothalamic synapses on the function of the thalamus is not well understood because thalamic neurones have complex intrinsic firing properties and interact through multiple types of synaptic receptors. Here we investigate these complex interactions using computational models. We show first, using models of reconstructed thalamic relay neurones, that the effect of corticothalamic synapses on relay cells can be similar to that of afferent synapses, in amplitude, kinetics and timing, although these synapses are located in different regions of the dendrites. This suggests that cortical EPSPs may complement (or predict) the afferent information. Second, using models of reconstructed thalamic reticular neurones, we show that high densities of the low-threshold Ca2+ current in dendrites can give these cells an exquisite sensitivity to cortical EPSPs, but only if their dendrites are hyperpolarized. This property has consequences at the level of thalamic circuits, where corticothalamic EPSPs evoke bursts in reticular neurones and recruit relay cells predominantly through feedforward inhibition. On the other hand, with depolarized dendrites, thalamic reticular neurones do not generate bursts and the cortical influence on relay cells is mostly excitatory. Models therefore suggest that the cortical influence can either promote or antagonize the relay of information, depending on the state of the dendrites of reticular neurones. The control of these dendrites may therefore be a determinant of attentional mechanisms. We also review the effect of corticothalamic feedback at the network level, and show how the cortical control over the thalamus is essential in co-ordinating widespread, coherent oscillations. We suggest mechanisms by which different modes of corticothalamic interaction would allow oscillations of very different spatiotemporal coherence to coexist in the thalamocortical system.     

Irregular dynamics in up and down cortical states

Complex coherent dynamics is present in a wide variety of neural systems. A typical example is the voltage transitions between up and down states observed in cortical areas in the brain. In this work, we study this phenomenon via a biologically motivated stochastic model of up and down transitions. The model is constituted by a simple bistable rate dynamics, where the synaptic current is modulated by short-term synaptic processes which introduce stochasticity and temporal correlations. A complete analysis of our model, both with mean-field approaches and numerical simulations, shows the appearance of complex transitions between high (up) and low (down) neural activity states, driven by the synaptic noise, with permanence times in the up state distributed according to a power-law. We show that the experimentally observed large fluctuation in up and down permanence times can be explained as the result of sufficiently noisy dynamical synapses with sufficiently large recovery times. Static synapses cannot account for this behavior, nor can dynamical synapses in the absence of noise.     

An artificial hysteresis binary neuron: a model suppressing the oscillatory behaviors of neural dynamics

A hysteresis binary McCulloch-Pitts neuron model is proposed in order to suppress the complicated oscillatory behaviors of neural dynamics. The artificial hysteresis binary neural network is used for scheduling time-multiplex crossbar switches in order to demonstrate the effects of hysteresis. Time-multiplex crossbar switching systems must control traffic on demand such that packet blocking probability and packet waiting time are minimized. The system using n×n processing elements solves an n×n crossbar-control problem with O(1) time, while the best existing parallel algorithm requires O(n) time. The hysteresis binary neural network maximizes the throughput of packets through a crossbar switch. The solution quality of our system does not degrade with the problem size.     

Spatial pattern analysis in forest dynamics: deviation from power law and direction of regeneration waves

Grid-based models have been used to understand spatial heterogeneity of the vegetation height in forests and to analyze spatio-temporal dynamics of the forest regeneration process. In this report, we present two methods of identifying lattice models when spatio-temporal data are given. The first method detects directionality of regeneration waves based on the timing of local disturbance events. The second evaluates the forest pattern by recording the fraction of high and low vegetation areas at multiple spatial scales. We illustrate these methods by applying them to patterns generated using three simple stochastic lattice models: (1) two-state model, distinguishing sites with high and low vegetation, (2) three-state model, in which sites can be in an additional disturbed state, and (3) Shimagare model, which considers a continuous range of states. The combination of the two methods provides efficient means of distinguishing the models. The first method has a more direct ecological meaning, while the second is useful when forest data are limited in time.     

Modulation of the dynamics of cerebellar Purkinje cells through the interaction of excitatory and inhibitory feedforward pathways

The dynamics of cerebellar neuronal networks is controlled by the underlying building blocks of neurons and synapses between them. For which, the computation of Purkinje cells (PCs), the only output cells of the cerebellar cortex, is implemented through various types of neural pathways interactively routing excitation and inhibition converged to PCs. Such tuning of excitation and inhibition, coming from the gating of specific pathways as well as short-term plasticity (STP) of the synapses, plays a dominant role in controlling the PC dynamics in terms of firing rate and spike timing. PCs receive cascade feedforward inputs from two major neural pathways: the first one is the feedforward excitatory pathway from granule cells (GCs) to PCs; the second one is the feedforward inhibition pathway from GCs, via molecular layer interneurons (MLIs), to PCs. The GC-PC pathway, together with short-term dynamics of excitatory synapses, has been a focus over past decades, whereas recent experimental evidence shows that MLIs also greatly contribute to controlling PC activity. Therefore, it is expected that the diversity of excitation gated by STP of GC-PC synapses, modulated by strong inhibition from MLI-PC synapses, can promote the computation performed by PCs. However, it remains unclear how these two neural pathways are interacted to modulate PC dynamics. Here using a computational model of PC network installed with these two neural pathways, we addressed this question to investigate the change of PC firing dynamics at the level of single cell and network. We show that the nonlinear characteristics of excitatory STP dynamics can significantly modulate PC spiking dynamics mediated by inhibition. The changes in PC firing rate, firing phase, and temporal spike pattern, are strongly modulated by these two factors in different ways. MLIs mainly contribute to variable delays in the postsynaptic action potentials of PCs while modulated by excitation STP. Notably, the diversity of synchronization and pause response in the PC network is governed not only by the balance of excitation and inhibition, but also by the synaptic STP, depending on input burst patterns. Especially, the pause response shown in the PC network can only emerge with the interaction of both pathways. Together with other recent findings, our results show that the interaction of feedforward pathways of excitation and inhibition, incorporated with synaptic short-term dynamics, can dramatically regulate the PC activities that consequently change the network dynamics of the cerebellar circuit.     

Memory Capacity of Networks with Stochastic Binary Synapses

In standard attractor neural network models, specific patterns of activity are stored in the synaptic matrix, so that they become fixed point attractors of the network dynamics. The storage capacity of such networks has been quantified in two ways: the maximal number of patterns that can be stored, and the stored information measured in bits per synapse. In this paper, we compute both quantities in fully connected networks of N binary neurons with binary synapses, storing patterns with coding level , in the large and sparse coding limits (). We also derive finite-size corrections that accurately reproduce the results of simulations in networks of tens of thousands of neurons. These methods are applied to three different scenarios: (1) the classic Willshaw model, (2) networks with stochastic learning in which patterns are shown only once (one shot learning), (3) networks with stochastic learning in which patterns are shown multiple times. The storage capacities are optimized over network parameters, which allows us to compare the performance of the different models. We show that finite-size effects strongly reduce the capacity, even for networks of realistic sizes. We discuss the implications of these results for memory storage in the hippocampus and cerebral cortex.     

Deformation of Attractor Landscape via Cholinergic Presynaptic Modulations: A Computational Study Using a Phase Neuron Model

Corticopetal acetylcholine (ACh) is released transiently from the nucleus basalis of Meynert (NBM) into the cortical layers and is associated with top-down attention. Recent experimental data suggest that this release of ACh disinhibits layer 2/3 pyramidal neurons (PYRs) via muscarinic presynaptic effects on inhibitory synapses. Together with other possible presynaptic cholinergic effects on excitatory synapses, this may result in dynamic and temporal modifications of synapses associated with top-down attention. However, the system-level consequences and cognitive relevance of such disinhibitions are poorly understood. Herein, we propose a theoretical possibility that such transient modifications of connectivity associated with ACh release, in addition to top-down glutamatergic input, may provide a neural mechanism for the temporal reactivation of attractors as neural correlates of memories. With baseline levels of ACh, the brain returns to quasi-attractor states, exhibiting transitive dynamics between several intrinsic internal states. This suggests that top-down attention may cause the attention-induced deformations between two types of attractor landscapes: the quasi-attractor landscape (Q-landscape, present under low-ACh, non-attentional conditions) and the attractor landscape (A-landscape, present under high-ACh, top-down attentional conditions). We present a conceptual computational model based on experimental knowledge of the structure of PYRs and interneurons (INs) in cortical layers 1 and 2/3 and discuss the possible physiological implications of our results.     

Host-parasitoid spatial dynamics in heterogeneous landscapes

This paper explores the effect of spatial processes in a heterogeneous environment on the dynamics of a host-parasitoid interaction. The environment consists of a lattice of favourable (habitat) and hostile (matrix) hexagonal cells, whose spatial distribution is measured by habitat proportion and spatial autocorrelation (inverse of fragmentation). At each time step, a fixed fraction of both populations disperses to the adjacent cells where it reproduces following the Nicholson-Bailey model. Aspects of the dynamics analysed include extinction, stability, cycle period and amplitude, and the spatial patterns emerging from the dynamics.
We find that, depending primarily on the fraction of the host population that disperses in each generation and on the landscape geometry, five classes of spatio-temporal dynamics can be objectively distinguished: spatial chaos, spirals, metapopulation, mainland-island and spiral fragments. The first two are commonly found in theoretical studies of homogeneous landscapes. The other three are direct consequences of the heterogeneity and have strong similarities to dynamic patterns observed in real systems (e.g. extinction-recolonisation, source-sink, outbreaks, spreading waves).
We discuss the processes that generate these patterns and allow the system to persist. The importance of these results is threefold: first, our model merges into a same theoretical framework dynamics commonly observed in the field that are usually modelled independently. Second, these dynamics and patterns are explained by dispersal rate and common landscape statistics, thus linking in a practical way population ecology to landscape ecology. Third, we show that the landscape geometry has a qualitative effect on the length of the cycles and, in particular, we demonstrate how very long periods can be produced by spatial processes.     

On the dynamics of electrically-coupled neurons with inhibitory synapses

We study the dynamics and bifurcations of noise-free neurons coupled by gap junctions and inhibitory synapses, using both delayed delta functions and alpha functions to model the latter. We focus on the case of two cells, as in the studies of Chow and Kopell (2000) and Lewis and Rinzel (2003), but also show that stable asynchronous splay states exist for globally coupled networks of N cells dominated by subthreshold electrical coupling. Our results agree with those of Lewis and Rinzel (2003) in the weak coupling range, but our Poincaré map analysis yields more information about global behavior and domains of attraction, and we show that the explicit discontinuous maps derived using delayed delta functions compare well with the continuous history-dependent, implicitly-defined maps derived from alpha functions. We find that increased bias currents, super-threshold electrical coupling and synaptic delays promote synchrony, while sub-threshold electrical coupling and fast synapses promote asynchrony. We compare our analytical results with simulations of an ionic current model of spiking cells, and briefly discuss implications for stimulus response modes of locus coeruleus and for central pattern generators. Action Editor: F. Skinner     

Multiscale brain modelling

A central difficulty of brain modelling is to span the range of spatio-temporal scales from synapses to the whole brain. This paper overviews results from a recent model of the generation of brain electrical activity that incorporates both basic microscopic neurophysiology and large-scale brain anatomy to predict brain electrical activity at scales from a few tenths of a millimetre to the whole brain. This model incorporates synaptic and dendritic dynamics, nonlinearity of the firing response, axonal propagation and corticocortical and corticothalamic pathways. Its relatively few parameters measure quantities such as synaptic strengths, corticothalamic delays, synaptic and dendritic time constants, and axonal ranges, and are all constrained by independent physiological measurements. It reproduces quantitative forms of electroencephalograms seen in various states of arousal, evoked response potentials, coherence functions, seizure dynamics and other phenomena. Fitting model predictions to experimental data enables underlying physiological parameters to be inferred, giving a new non-invasive window into brain function that complements slower, but finer-resolution, techniques such as fMRI. Because the parameters measure physiological quantities relating to multiple scales, and probe deep structures such as the thalamus, this will permit the testing of a range of hypotheses about vigilance, cognition, drug action and brain function. In addition, referencing to a standardized database of subjects adds strength and specificity to characterizations obtained.     

Capacity analysis in multi-state synaptic models: a retrieval probability perspective

We define the memory capacity of networks of binary neurons with finite-state synapses in terms of retrieval probabilities of learned patterns under standard asynchronous dynamics with a predetermined threshold. The threshold is set to control the proportion of non-selective neurons that fire. An optimal inhibition level is chosen to stabilize network behavior. For any local learning rule we provide a computationally efficient and highly accurate approximation to the retrieval probability of a pattern as a function of its age. The method is applied to the sequential models (Fusi and Abbott, Nat Neurosci 10:485–493, 2007) and meta-plasticity models (Fusi et al., Neuron 45(4):599–611, 2005; Leibold and Kempter, Cereb Cortex 18:67–77, 2008). We show that as the number of synaptic states increases, the capacity, as defined here, either plateaus or decreases. In the few cases where multi-state models exceed the capacity of binary synapse models the improvement is small.     

Analysis of multidimensional neural activity via CNN-UM

In this paper we show that the Cellular Nonlinear Network Universal Machine (CNN-UM) is an excellent tool for analyzing time series of multidimensional binary signals. The developed algorithm is dedicated to process electrophysiological multi-neuron recordings: our aim is to find specific multidimensional activity patterns, which may reflect higher order functional cell-assemblies. The analysis consists of two parts: first, the occurrences of different patterns are counted, then the statistical significance of each occurrence frequency is calculated separately.     

Spatial self-organization in a cyclic resource-species model

Biological communities are remarkable in their ability to form cooperative ensembles that lead to coexistence through various types of niche partitioning, usually intimately tied to spatial structure. This is especially true in microbial settings where differential expression and regulation of genes allows members of a given species to alter their lifestyle so as to fill a functional role within the community. The resulting species interactions can involve feedback, as in the case of some bacterial consortia that participate in the cooperative degradation of a given resource in a succession of steps and in such a way that certain "later" species provide catalytic support for the primary degrader. We seek to capture the essential features of such spatially extended biological systems by introducing a lattice-based stochastic spatial model (interacting particle system) with cyclic local dynamics. Here, a given site progresses through a sequence of resource and species states in a prescribed order. Furthermore, this succession of states (at a site) is assumed to form a cyclic pattern due to a natural feedback mechanism. We explore conditions under which all the species are able to coexist and consider the extent to which this coexistence requires the development of spatio-temporal patterns, including spiral waves. This self-organization, if it occurs, results when synchronization of the dynamics at the microscopic level leads to macroscopic patterns. These patterns result in consumer-driven resource fluctuations that generate a form of spatio-temporal niche partitioning. As with most models of this complexity, we employ a mixture of mathematical analysis and simulations to develop an understanding of the resulting dynamics.     

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.     

Mathematical models of predator/prey/plant interactions in a patch environment

Several tritrophic systems are characterized by local over-exploitation of the food source. Interactions between predatory mites, spider mites and their host plants are an example of such systems: either the spider mites over-exploit local patches of host plants or the spider mites are exterminated by predatory mites. It is often stated that modelling the overall population dynamics of such systems in a realistic way would soon lead to an unmanageable edifice. We advocate, however, the use of physiologically structured population models as a both general and formal mathematical framework. The advantage is that analytically tractable models may be obtained from the complex ‘master’ model by time-scale arguments or special choices of model ingredients. In this way a network of models can be derived, each concentrating on a particular aspect, all inadequate to cover the entire spectrum, but together (we hope) providing a coherent set of insights the relative importance of which can be assessed by computer experiments on the ‘master’ model. In this paper a rather realistic model of predator/prey interactions in an ensemble of host-plant patches is presented and, as an example of our approach, some special cases are derived from that model. Their analysis provided some first, useful insights. It is shown that prolonged duration of the prey-dispersal phase and prey dispersal from predator (-invaded prey) patches may result in a stable steady state, whereas a humped plant-production function may — under certain conditions — result in two stable steady states.