Identifying Crucial Parameter Correlations Maintaining Bursting Activity

Recent experimental and computational studies suggest that linearly correlated sets of parameters (intrinsic and synaptic properties of neurons) allow central pattern-generating networks to produce and maintain their rhythmic activity regardless of changing internal and external conditions. To determine the role of correlated conductances in the robust maintenance of functional bursting activity, we used our existing database of half-center oscillator (HCO) model instances of the leech heartbeat CPG. From the database, we identified functional activity groups of burster (isolated neuron) and half-center oscillator model instances and realistic subgroups of each that showed burst characteristics (principally period and spike frequency) similar to the animal. To find linear correlations among the conductance parameters maintaining functional leech bursting activity, we applied Principal Component Analysis (PCA) to each of these four groups. PCA identified a set of three maximal conductances (leak current, _Leak; a persistent K current, _K2; and of a persistent Na+ current, _P) that correlate linearly for the two groups of burster instances but not for the HCO groups. Visualizations of HCO instances in a reduced space suggested that there might be non-linear relationships between these parameters for these instances. Experimental studies have shown that period is a key attribute influenced by modulatory inputs and temperature variations in heart interneurons. Thus, we explored the sensitivity of period to changes in maximal conductances of _Leak, _K2, and _P, and we found that for our realistic bursters the effect of these parameters on period could not be assessed because when varied individually bursting activity was not maintained.     

Leader neurons in leaky integrate and fire neural network simulations

In this paper, we highlight the topological properties of leader neurons whose existence is an experimental fact. Several experimental studies show the existence of leader neurons in population bursts of activity in 2D living neural networks (Eytan and Marom, J Neurosci 26(33):8465–8476, 2006; Eckmann et al., New J Phys 10(015011), 2008). A leader neuron is defined as a neuron which fires at the beginning of a burst (respectively network spike) more often than we expect by chance considering its mean firing rate. This means that leader neurons have some burst triggering power beyond a chance-level statistical effect. In this study, we characterize these leader neuron properties. This naturally leads us to simulate neural 2D networks. To build our simulations, we choose the leaky integrate and fire (lIF) neuron model (Gerstner and Kistler 2002; Cessac, J Math Biol 56(3):311–345, 2008), which allows fast simulations (Izhikevich, IEEE Trans Neural Netw 15(5):1063–1070, 2004; Gerstner and Naud, Science 326:379–380, 2009). The dynamics of our lIF model has got stable leader neurons in the burst population that we simulate. These leader neurons are excitatory neurons and have a low membrane potential firing threshold. Except for these two first properties, the conditions required for a neuron to be a leader neuron are difficult to identify and seem to depend on several parameters involved in the simulations themselves. However, a detailed linear analysis shows a trend of the properties required for a neuron to be a leader neuron. Our main finding is: A leader neuron sends signals to many excitatory neurons as well as to few inhibitory neurons and a leader neuron receives only signals from few other excitatory neurons. Our linear analysis exhibits five essential properties of leader neurons each with different relative importance. This means that considering a given neural network with a fixed mean number of connections per neuron, our analysis gives us a way of predicting which neuron is a good leader neuron and which is not. Our prediction formula correctly assesses leadership for at least ninety percent of neurons.     

High Prevalence of Multistability of Rest States and Bursting in a Database of a Model Neuron

Flexibility in neuronal circuits has its roots in the dynamical richness of their neurons. Depending on their membrane properties single neurons can produce a plethora of activity regimes including silence, spiking and bursting. What is less appreciated is that these regimes can coexist with each other so that a transient stimulus can cause persistent change in the activity of a given neuron. Such multistability of the neuronal dynamics has been shown in a variety of neurons under different modulatory conditions. It can play either a functional role or present a substrate for dynamical diseases. We considered a database of an isolated leech heart interneuron model that can display silent, tonic spiking and bursting regimes. We analyzed only the cases of endogenous bursters producing functional half-center oscillators (HCOs). Using a one parameter (the leak conductance ()) bifurcation analysis, we extended the database to include silent regimes (stationary states) and systematically classified cases for the coexistence of silent and bursting regimes. We showed that different cases could exhibit two stable depolarized stationary states and two hyperpolarized stationary states in addition to various spiking and bursting regimes. We analyzed all cases of endogenous bursters and found that 18% of the cases were multistable, exhibiting coexistences of stationary states and bursting. Moreover, 91% of the cases exhibited multistability in some range of . We also explored HCOs built of multistable neuron cases with coexisting stationary states and a bursting regime. In 96% of cases analyzed, the HCOs resumed normal alternating bursting after one of the neurons was reset to a stationary state, proving themselves robust against this perturbation.     

Reliable circuits from irregular neurons: A dynamical approach to understanding central pattern generators

Central pattern generating neurons from the lobster stomatogastric ganglion were analyzed using new nonlinear methods. The LP neuron was found to have only four or five degrees of freedom in the isolated condition and displayed chaotic behavior. We show that this chaotic behavior could be regularized by periodic pulses of negative current injected into the neuron or by coupling it to another neuron via inhibitory connections. We used both a modified Hindmarsh-Rose model to simulate the neurons behavior phenomenologically and a more realistic conductance-based model so that the modeling could be linked to the experimental observations. Both models were able to capture the dynamics of the neuron behavior better than previous models. We used the Hindmarsh-Rose model as the basis for building electronic neurons which could then be integrated into the biological circuitry. Such neurons were able to rescue patterns which had been disabled by removing key biological neurons from the circuit.     

Bursts and isolated spikes code for opposite movement directions in midbrain electrosensory neurons

Directional selectivity, in which neurons respond strongly to an object moving in a given direction but weakly or not at all to the same object moving in the opposite direction, is a crucial computation that is thought to provide a neural correlate of motion perception. However, directional selectivity has been traditionally quantified by using the full spike train, which does not take into account particular action potential patterns. We investigated how different action potential patterns, namely bursts (i.e. packets of action potentials followed by quiescence) and isolated spikes, contribute to movement direction coding in a mathematical model of midbrain electrosensory neurons. We found that bursts and isolated spikes could be selectively elicited when the same object moved in opposite directions. In particular, it was possible to find parameter values for which our model neuron did not display directional selectivity when the full spike train was considered but displayed strong directional selectivity when bursts or isolated spikes were instead considered. Further analysis of our model revealed that an intrinsic burst mechanism based on subthreshold T-type calcium channels was not required to observe parameter regimes for which bursts and isolated spikes code for opposite movement directions. However, this burst mechanism enhanced the range of parameter values for which such regimes were observed. Experimental recordings from midbrain neurons confirmed our modeling prediction that bursts and isolated spikes can indeed code for opposite movement directions. Finally, we quantified the performance of a plausible neural circuit and found that it could respond more or less selectively to isolated spikes for a wide range of parameter values when compared with an interspike interval threshold. Our results thus show for the first time that different action potential patterns can differentially encode movement and that traditional measures of directional selectivity need to be revised in such cases.     

First return maps for the dynamics of synaptically coupled conditional bursters

The pre-Bötzinger complex (preBötc) in the mammalian brainstem has an important role in generating respiratory rhythms. An influential differential equation model for the activity of individual neurons in the preBötc yields transitions from quiescence to bursting to tonic spiking as a parameter is varied. Further, past work has established that bursting dynamics can arise from a pair of tonic model cells coupled with synaptic excitation. In this paper, we analytically derive one- and two-dimensional maps from the differential equations for a self-coupled neuron and a two-neuron network, respectively. Using a combination of analysis and simulations of these maps, we explore the possible forms of dynamics that the model networks can produce as well as which transitions between dynamic regimes are mathematically possible.     

Excitability of the soma in central nervous system neurons

The ability of the soma of a spinal dorsal horn neuron, a spinal ventral horn neuron (presumably a motoneuron), and a hippocampal pyramidal neuron to generate action potentials was studied using patch-clamp recordings from rat spinal cord slices, the "entire soma isolation" method, and computer simulations. By comparing original recordings from an isolated soma of a dorsal horn neuron with simulated responses, it was shown that computer models can be adequate for the study of somatic excitability. The modeled somata of both spinal neurons were unable to generate action potentials, showing only passive and local responses to current injections. A four- to eightfold increase in the original density of Na(+) channels was necessary to make the modeled somata of both spinal neurons excitable. In contrast to spinal neurons, the modeled soma of the hippocampal pyramidal neuron generated spikes with an overshoot of +9 mV. It is concluded that the somata of spinal neurons cannot generate action potentials and seem to resist their propagation from the axon to dendrites. In contrast, the soma of the hippocampal pyramidal neuron is able to generate spikes. It cannot initiate action potentials in the intact neurons, but it can support their back-propagation from the axon initial segment to dendrites.     

Computational modeling of synchronization process of the circadian timing system of mammals

This paper presents a model for the circadian temporization system of mammals which associates the synchronization dynamics of coupling oscillators to a set of equations able to reproduce the synaptic characteristics of somatodendritic membrane of neurons. The circadian timing system is organized in a way to receive information from the external and internal environments, and its function is the timing organization of physiological and behavioral processes in a circadian pattern. Circadian timing system in mammals is constituted by a group of structures which includes the suprachiasmatic nucleus, the intergeniculate leaflet and the pineal gland. In suprachiasmatic nucleus are found neuron groups working as a biological pacemaker—the so-called biological master clock. By means of numerical simulations using the Kuramoto model, we simulated the dynamics behavior of the biological pacemaker. For this we used a set of 1,000 coupled oscillators with long-range coupling, which were distributed on a 10 × 10 × 10 spherical lattice, and a new method to estimate the order parameter, which characterizes the degree of synchronization of oscillator system. Our model has been able to produce frequency responses in accordance with physiological patterns, and to reproduce two fundamental characteristics of biological rhythms: the endogenous generation and synchronization to the light–dark cycle.     

Disentanglement of local field potential sources by independent component analysis

The spontaneous activity of working neurons yields synaptic currents that mix up in the volume conductor. This activity is picked up by intracerebral recording electrodes as local field potentials (LFPs), but their separation into original informative sources is an unresolved problem. Assuming that synaptic currents have stationary placing we implemented independent component model for blind source separation of LFPs in the hippocampal CA1 region. After suppressing contaminating sources from adjacent regions we obtained three main local LFP generators. The specificity of the information contained in isolated generators is much higher than in raw potentials as revealed by stronger phase-spike correlation with local putative interneurons. The spatial distribution of the population synaptic input corresponding to each isolated generator was disclosed by current-source density analysis of spatial weights. The found generators match with axonal terminal fields from subtypes of local interneurons and associational fibers from nearby subfields. The found distributions of synaptic currents were employed in a computational model to reconstruct spontaneous LFPs. The phase-spike correlations of simulated units and LFPs show laminar dependency that reflects the nature and magnitude of the synaptic currents in the targeted pyramidal cells. We propose that each isolated generator captures the synaptic activity driven by a different neuron subpopulation. This offers experimentally justified model of local circuits creating extracellular potential, which involves distinct neuron subtypes.     

Nonlinear electronic circuit with neuron like bursting and spiking dynamics

It is difficult to design electronic nonlinear devices capable of reproducing complex oscillations because of the lack of general constructive rules, and because of stability problems related to the dynamical robustness of the circuits. This is particularly true for current analog electronic circuits that implement mathematical models of bursting and spiking neurons. Here we describe a novel, four-dimensional and dynamically robust nonlinear analog electronic circuit that is intrinsic excitable, and that displays frequency adaptation bursting and spiking oscillations. Despite differences from the classical Hodgkin–Huxley (HH) neuron model, its bifurcation sequences and dynamical properties are preserved, validating the circuit as a neuron model. The circuit's performance is based on a nonlinear interaction of fast–slow circuit blocks that can be clearly dissected, elucidating burst's starting, sustaining and stopping mechanisms, which may also operate in real neurons. Our analog circuit unit is easily linked and may be useful in building networks that perform in real-time.     

Entrainment, Instability, Quasi-periodicity, and Chaos in a Compound Neural Oscillator

We studied the dynamical behavior of a class of compound central pattern generator (CPG) models consisting of a simple neural network oscillator driven by both constant and periodic inputs of varying amplitudes, frequencies, and phases. We focused on a specific oscillator composed of two mutually inhibiting types of neuron (inspiratory and expiratory neurons) that may be considered as a minimal model of the mammalian respiratory rhythm generator. The simulation results demonstrated how a simple CPG model— with a minimum number of neurons and mild nonlinearities— may reproduce a host of complex dynamical behaviors under various periodic inputs. In particular, the network oscillated spontaneously only when both neurons received adequate and proportionate constant excitations. In the presence of a periodic source, the spontaneous rhythm was overriden by an entrained oscillation of varying forms depending on the nature of the source. Stable entrained oscillations were inducible by two types of inputs: (1) anti-phase periodic inputs with alternating agonist-antagonist drives to both neurons and (2) a single periodic drive to only one of the neurons. In-phase inputs, which exert periodic drives of similar magnitude and phase relationships to both neurons, resulted in varying disruptions of the entrained oscillations including magnitude attenuation, harmonic and phase distortions, and quasi-periodic interference. In the absence of significant phasic feedback, chaotic motion developed only when the CPG was driven by multiple periodic inputs. Apneic episodes with repetitive alternation of active (intrinsic oscillation) and inactive (cessation of oscillation) states developed when the network was driven by a moderate periodic input of low frequency. %and amplitudes of intermediate strength, Similar results were demonstrated in other, more complex oscillator models (that is, half-center oscillator and three-phase respiratory network model). These theoretical results may have important implications in elucidating the mechanisms of rhythmogenesis in the mature and developing respiratory CPG as well as other compound CPGs in mammalian and invertebrate nervous systems.     

Heuristics for the Hodgkin–Huxley system

Hodgkin and Huxley (HH) discovered that voltages control ionic currents in nerve membranes. This led them to describe electrical activity in a neuronal membrane patch in terms of an electronic circuit whose characteristics were determined using empirical data. Due to the complexity of this model, a variety of heuristics, including relaxation oscillator circuits and integrate-and-fire models, have been used to investigate activity in neurons, and these simpler models have been successful in suggesting experiments and explaining observations. Connections between most of the simpler models had not been made clear until recently. Shown here are connections between these heuristics and the full HH model. In particular, we study a new model (Type III circuit): It includes the van der Pol-based models; it can be approximated by a simple integrate-and-fire model; and it creates voltages and currents that correspond, respectively, to the h and V components of the HH system.     

Energy efficiency of information transmission by electrically coupled neurons

The generation of spikes by neurons is energetically a costly process. This paper studies the consumption of energy and the information entropy in the signalling activity of a model neuron both when it is supposed isolated and when it is coupled to another neuron by an electrical synapse. The neuron has been modelled by a four-dimensional Hindmarsh–Rose type kinetic model for which an energy function has been deduced. For the isolated neuron values of energy consumption and information entropy at different signalling regimes have been computed. For two neurons coupled by a gap junction we have analyzed the roles of the membrane and synapse in the contribution of the energy that is required for their organized signalling. Computational results are provided for cases of identical and nonidentical neurons coupled by unidirectional and bidirectional gap junctions. One relevant result is that there are values of the coupling strength at which the organized signalling of two neurons induced by the gap junction takes place at relatively low values of energy consumption and the ratio of mutual information to energy consumption is relatively high. Therefore, communicating at these coupling values could be energetically the most efficient option.     

Fictive chewing activity in motor neurons and interneurons of the suboesophageal ganglion of Manduca sexta larvae

Alternating antiphasic rhythmic activity was observed in opener and closer mandibular motor neurons in the isolated suboesophageal ganglion of the larva of Manduca sexta (Lepidoptera: Sphingidae). This was interpreted provisionally as fictive chewing; the pattern is similar to that seen in semiintact animals but of lower frequency. Additionally, a variety of associated rhythmic activities were observed in suboesophageal interneurons. These could be classified into several different physiological types by their activity patterns in relation to the chewing cycle. Some of these neurons can modulate the rhythm when injected with current. It seems likely that they are part of or associated with a central pattern generator circuit for chewing.Abbreviations A anterior - CEC circumoesophageal connective - Cl-MN closer motor neuron - IN interneuron - MdN mandibular nerve - MN motor neuron - O-MN opener motor neuron     

Using interspike intervals to quantify noise effects on spike trains in temperature encoding neurons

This paper examines how noise interacts with the non-linear dynamical mechanisms of neuronal stimulus. We study the spike trains generated by a minimal Hodgkin-Huxley type model of a cold receptor neuron. The distributions of interspike intervals(ISIs) of purely deterministic simulations exhibit considerable differences compared to the noisy ones. We quantify the effect of noise using ISI return plots and the ISI-distance recently proposed by Kreuz et al. (J Neurosci Meth, 165:151–161, 2007). It is shown that the spike trains of a cold receptor neuron are more strongly affected by noise for low temperatures than for high temperatures. This trend is also observed in both regimes of cold receptors: tonic firing(which occurs for low and high temperatures) and bursting (which occurs for intermediate temperatures).     

Oscillations in Large-Scale Cortical Networks: Map-Based Model

We develop a new computationally efficient approach for the analysis of complex large-scale neurobiological networks. Its key element is the use of a new phenomenological model of a neuron capable of replicating important spike pattern characteristics and designed in the form of a system of difference equations (a map). We developed a set of map-based models that replicate spiking activity of cortical fast spiking, regular spiking and intrinsically bursting neurons. Interconnected with synaptic currents these model neurons demonstrated responses very similar to those found with Hodgkin-Huxley models and in experiments. We illustrate the efficacy of this approach in simulations of one- and two-dimensional cortical network models consisting of regular spiking neurons and fast spiking interneurons to model sleep and activated states of the thalamocortical system. Our study suggests that map-based models can be widely used for large-scale simulations and that such models are especially useful for tasks where the modeling of specific firing patterns of different cell classes is important.     

Probing the dynamics of identified neurons with a data-driven modeling approach

In controlling animal behavior the nervous system has to perform within the operational limits set by the requirements of each specific behavior. The implications for the corresponding range of suitable network, single neuron, and ion channel properties have remained elusive. In this article we approach the question of how well-constrained properties of neuronal systems may be on the neuronal level. We used large data sets of the activity of isolated invertebrate identified cells and built an accurate conductance-based model for this cell type using customized automated parameter estimation techniques. By direct inspection of the data we found that the variability of the neurons is larger when they are isolated from the circuit than when in the intact system. Furthermore, the responses of the neurons to perturbations appear to be more consistent than their autonomous behavior under stationary conditions. In the developed model, the constraints on different parameters that enforce appropriate model dynamics vary widely from some very tightly controlled parameters to others that are almost arbitrary. The model also allows predictions for the effect of blocking selected ionic currents and to prove that the origin of irregular dynamics in the neuron model is proper chaoticity and that this chaoticity is typical in an appropriate sense. Our results indicate that data driven models are useful tools for the in-depth analysis of neuronal dynamics. The better consistency of responses to perturbations, in the real neurons as well as in the model, suggests a paradigm shift away from measuring autonomous dynamics alone towards protocols of controlled perturbations. Our predictions for the impact of channel blockers on the neuronal dynamics and the proof of chaoticity underscore the wide scope of our approach.     

Exploring neuronal bistability at the depolarization block

Many neurons display bistability-coexistence of two firing modes such as bursting and tonic spiking or tonic spiking and silence. Bistability has been proposed to endow neurons with richer forms of information processing in general and to be involved in short-term memory in particular by allowing a brief signal to elicit long-lasting changes in firing. In this paper, we focus on bistability that allows for a choice between tonic spiking and depolarization block in a wide range of the depolarization levels. We consider the spike-producing currents in two neurons, models of which differ by the parameter values. Our dopaminergic neuron model displays bistability in a wide range of applied currents at the depolarization block. The Hodgkin-Huxley model of the squid giant axon shows no bistability. We varied parameter values for the model to analyze transitions between the two parameter sets. We show that bistability primarily characterizes the inactivation of the Na(+) current. Our study suggests a connection between the amount of the Na(+) window current and the length of the bistability range. For the dopaminergic neuron we hypothesize that bistability can be linked to a prolonged action of antipsychotic drugs.     

How do glutamatergic and GABAergic cells contribute to synchronization in the medial septum?

The medial septum-diagonal band (MSDB) complex is considered as a pacemaker for the hippocampal theta rhythm. Identification of the different cell types, their electro-physiological properties and their possible function in the generation of a synchronized activity in the MSDB is a hot topic. A recent electro-physiological study showed the presence of two antiphasically firing populations of parvalbumin containing GABAergic neurons in the MSDB. Other papers described a network of cluster-firing glutamatergic neurons, which is able to generate synchronized activity in the MSDB. We propose two different computer models for the generation of synchronized population theta oscillation in the MSDB and compare their properties. In the first model GABAergic neurons are intrinsically theta periodic cluster-firing cells; while in the second model GABAergic cells are fast-firing cells and receive periodic input from local glutamatergic neurons simulated as cluster-firing cells. Using computer simulations we show that the GABAergic neurons in both models are capable of generating antiphasic theta periodic population oscillation relying on local, septal mechanisms. In the first model antiphasic theta synchrony could emerge if GABAergic neurons form two populations preferentially innervate each other. In the second model in-phase synchronization of glutamatergic neurons does not require specific network structure, and the network of these cells are able to act as a theta pacemaker for the local fast-firing GABAergic circuit. Our simulations also suggest that neurons being non-cluster-firing in vitro might exhibit clustering properties when connected into a network in vivo. Action Editor: David Golomb     

Oscillations and Synchrony in Large-scale Cortical Network Models

Intrinsic neuronal and circuit properties control the responses of large ensembles of neurons by creating spatiotemporal patterns of activity that are used for sensory processing, memory formation, and other cognitive tasks. The modeling of such systems requires computationally efficient single-neuron models capable of displaying realistic response properties. We developed a set of reduced models based on difference equations (map-based models) to simulate the intrinsic dynamics of biological neurons. These phenomenological models were designed to capture the main response properties of specific types of neurons while ensuring realistic model behavior across a sufficient dynamic range of inputs. This approach allows for fast simulations and efficient parameter space analysis of networks containing hundreds of thousands of neurons of different types using a conventional workstation. Drawing on results obtained using large-scale networks of map-based neurons, we discuss spatiotemporal cortical network dynamics as a function of parameters that affect synaptic interactions and intrinsic states of the neurons.