Period doubling of calcium spike firing in a model of a Purkinje cell dendrite

Emergent excitability in populations of nonexcitable units

Population bursts in a large ensemble of coupled elements result from the interplay between the local excitable properties of the nodes and the global network topology. Here, collective excitability and self-sustained bursting oscillations are shown to spontaneously emerge in globally coupled populations of nonexcitable units subject to adaptive coupling. The ingredients to observe collective excitability are the coexistence of states with different degrees of synchronization joined to a global feedback acting, on a slow timescale, against the synchronization (desynchronization) of the oscillators. These regimes are illustrated for two paradigmatic classes of coupled rotators, namely, the Kuramoto model with and without inertia. For the bimodal Kuramoto model we analytically show that the macroscopic evolution originates from the existence of a critical manifold organizing the fast collective dynamics on a slow timescale. Our results provide evidence that adaptation can induce excitability by maintaining a network permanently out of equilibrium.

A basic bifurcation structure from bursting to spiking of injured nerve fibers in a two-dimensional parameter space

Two different bifurcation scenarios of firing patterns with decreasing extracellular calcium concentrations were observed in identical sciatic nerve fibers of a chronic constriction injury (CCI) model when the extracellular 4-aminopyridine concentrations were fixed at two different levels. Both processes proceeded from period-1 bursting to period-1 spiking via complex or simple processes. Multiple typical experimental examples manifested dynamics closely matching those simulated in a recently proposed 4-dimensional model to describe the nonlinear dynamics of the CCI model, which included most cases of the bifurcation scenarios. As the extracellular 4-aminopyridine concentrations is increased, the structure of the bifurcation scenario becomes more complex. The results provide a basic framework for identifying the relationships between different neural firing patterns and different bifurcation scenarios and for revealing the complex nonlinear dynamics of neural firing patterns. The potential roles of the basic bifurcation structures in identifying the information process mechanism are discussed.

Quantitative imaging with Fucci and mathematics to uncover temporal dynamics of cell cycle progression

Cell cycle progression is strictly coordinated to ensure proper tissue growth, development, and regeneration of multicellular organisms. Spatiotemporal visualization of cell cycle phases directly helps us to obtain a deeper understanding of controlled, multicellular, cell cycle progression. The fluorescent ubiquitination-based cell cycle indicator (Fucci) system allows us to monitor, in living cells, the G1 and the S/G2/M phases of the cell cycle in red and green fluorescent colors, respectively. Since the discovery of Fucci technology, it has found numerous applications in the characterization of the timing of cell cycle phase transitions under diverse conditions and various biological processes. However, due to the complexity of cell cycle dynamics, understanding of specific patterns of cell cycle progression is still far from complete. In order to tackle this issue, quantitative approaches combined with mathematical modeling seem to be essential. Here, we review several studies that attempted to integrate Fucci technology and mathematical models to obtain quantitative information regarding cell cycle regulatory patterns. Focusing on the technological development of utilizing mathematics to retrieve meaningful information from the Fucci producing data, we discuss how the combined methods advance a quantitative understanding of cell cycle regulation.

The 40-year history of modeling active dendrites in cerebellar Purkinje cells: emergence of the first single cell "community model"

The subject of the effects of the active properties of the Purkinje cell dendrite on neuronal function has been an active subject of study for more than 40 years. Somewhat unusually, some of these investigations, from the outset have involved an interacting combination of experimental and model-based techniques. This article recounts that 40-year history, and the view of the functional significance of the active properties of the Purkinje cell dendrite that has emerged. It specifically considers the emergence from these efforts of what is arguably the first single cell "community" model in neuroscience. The article also considers the implications of the development of this model for future studies of the complex properties of neuronal dendrites.

Period doubling induced by thermal noise amplification in genetic circuits

Rhythms of life are dictated by oscillations, which take place in a wide rage of biological scales. In bacteria, for example, oscillations have been proven to control many fundamental processes, ranging from gene expression to cell divisions. In genetic circuits, oscillations originate from elemental block such as autorepressors and toggle switches, which produce robust and noise-free cycles with well defined frequency. In some circumstances, the oscillation period of biological functions may double, thus generating bistable behaviors whose ultimate origin is at the basis of intense investigations. Motivated by brain studies, we here study an “elemental” genetic circuit, where a simple nonlinear process interacts with a noisy environment. In the proposed system, nonlinearity naturally arises from the mechanism of cooperative stability, which regulates the concentration of a protein produced during a transcription process. In this elemental model, bistability results from the coherent amplification of environmental fluctuations due to a stochastic resonance of nonlinear origin. This suggests that the period doubling observed in many biological functions might result from the intrinsic interplay between nonlinearity and thermal noise.

Dynamics of period-doubling bifurcation to chaos in the spontaneous neural firing patterns

Period-doubling bifurcation to chaos were discovered in spontaneous firings of Onchidium pacemaker neurons. In this paper, we provide three cases of bifurcation processes related to period-doubling bifurcation cascades to chaos observed in the spontaneous firing patterns recorded from an injured site of rat sciatic nerve as a pacemaker. Period-doubling bifurcation cascades to period-4 (π(2,2)) firstly, and then to chaos, at last to a periodicity, which can be period-5, period-4 (π(4)) and period-3, respectively, in different pacemakers. The three bifurcation processes are labeled as case I, II and III, respectively, manifesting procedures different to those of period-adding bifurcation. Higher-dimensional unstable periodic orbits (UPOs) can be detected in the chaos, built close relationships to the periodic firing patterns. Case III bifurcation process is similar to that discovered in the Onchidium pacemaker neurons and simulated in theoretical model-Chay model. The extra-large Feigenbaum constant manifesting in the period-doubling bifurcation process, induced by quasi-discontinuous characteristics exhibited in the first return maps of both ISI series and slow variable of Chay model, shows that higher-dimensional periodic behaviors appeared difficult within the period-doubling bifurcation cascades. The results not only provide examples of period-doubling bifurcation to chaos and chaos with higher-dimensional UPOs, but also reveal the dynamical features of the period-doubling bifurcation cascades to chaos.

THE BIFURCATION STRUCTURE OF FIRING PATTERN TRANSITIONS IN THE CHAY NEURONAL PACEMAKER MODEL

Different transitions of neuronal firing patterns are explored by the combination of experimental results, numerical simulation and bifurcation analysis.

Three types of firing sequences with respect to extracellular calcium concentration ([ Ca 2+] o ) were observed in experiments on neural pacemakers. In accordance with them, the corresponding transitions of neuronal firing patterns are surveyed by standard bifurcation analysis of the Chay model, where λn corresponds to different nerve fibers and VC is the dynamical parameter. The results are listed in this paper. Firstly, it is obtained that the transitions of periodic firing patterns from period-1 bursting to period-1 spiking without any bifurcation, from period-1 to -2 to -1 through a pair of period-doubling bifurcations and from period-1 to -2 to -1 through two pairs of period-doubling bifurcations. Secondly, one supercritical and two subcritical period-doubling bursting sequences with different appearances lead to chaos, respectively. Then the former transits directly to an inverse supercritical period-doubling spiking sequence via chaos, and the latter transit to it through the period-adding bursting sequences from period-1 to -3 and from -1 to -5 with chaotic bursting, respectively. Thirdly, we reveal the true nature of period-adding bursting sequence without chaotic bursting. Every periodic bursting is closely related to two period-doubling bifurcations of the corresponding periodic spiking, except for period-1 bursting appearing via Hopf bifurcation and disappearing via period-doubling bifurcation. As a consequence, the period-adding bursting sequence without chaotic bursting has a compound structure of elementary bifurcations with transitions from spiking to bursting. Thus period-adding bifurcation without chaos cannot be regarded as a new elementary bifurcation.

Mechanisms of firing patterns in fast-spiking cortical interneurons

Cortical fast-spiking (FS) interneurons display highly variable electrophysiological properties. Their spike responses to step currents occur almost immediately following the step onset or after a substantial delay, during which subthreshold oscillations are frequently observed. Their firing patterns include high-frequency tonic firing and rhythmic or irregular bursting (stuttering). What is the origin of this variability? In the present paper, we hypothesize that it emerges naturally if one assumes a continuous distribution of properties in a small set of active channels. To test this hypothesis, we construct a minimal, single-compartment conductance-based model of FS cells that includes transient Na⁺, delayed-rectifier K⁺, and slowly inactivating d-type K⁺ conductances. The model is analyzed using nonlinear dynamical system theory. For small Na⁺ window current, the neuron exhibits high-frequency tonic firing. At current threshold, the spike response is almost instantaneous for small d-current conductance, g_d, and it is delayed for larger g_d. As g_d further increases, the neuron stutters. Noise substantially reduces the delay duration and induces subthreshold oscillations. In contrast, when the Na⁺ window current is large, the neuron always fires tonically. Near threshold, the firing rates are low, and the delay to firing is only weakly sensitive to noise; subthreshold oscillations are not observed. We propose that the variability in the response of cortical FS neurons is a consequence of heterogeneities in their g_d and in the strength of their Na⁺ window current. We predict the existence of two types of firing patterns in FS neurons, differing in the sensitivity of the delay duration to noise, in the minimal firing rate of the tonic discharge, and in the existence of subthreshold oscillations. We report experimental results from intracellular recordings supporting this prediction.

About 25% of the neurons in the mammalian neocortex are inhibitory, namely reduce the activity of neurons they contact. These inhibitory neurons exhibit diversity of morphological, chemical, and biophysical properties, and their classification has recently been the focus of much debate. Even neurons belonging to a single class of “fast-spiking” (FS) display a large variety of firing patterns in response to standard square current pulses. Previous works proposed that this class is in fact a discrete set of neuronal subtypes with biophysical properties differing in a discontinuous way. In this work, we propose an alternative theory, according to which the biophysical properties of FS neurons are continuously distributed, but distinct firing patterns emerge due to highly nonlinear dynamics of these neurons. We ascertain this theory by exploring with mathematical techniques a biophysically based model of FS neurons. We demonstrate that variable firing responses of cortical FS neurons can be accounted for if one assumes heterogeneity in the strength of some of the ionic conductances underlying neuronal activity. Our theory predicts the existence of two main firing patterns of FS neurons. This prediction is verified by direct recordings in cortical slices.

Contribution of persistent Na+ current and M-type K+ current to somatic bursting in CA1 pyramidal cells: combined experimental and modeling study

The intrinsic firing modes of adult CA1 pyramidal cells vary along a continuum of "burstiness" from regular firing to rhythmic bursting, depending on the ionic composition of the extracellular milieu. Burstiness is low in neurons exposed to a normal extracellular Ca(2+) concentration ([Ca(2+)](o)), but is markedly enhanced by lowering [Ca(2+)](o), although not by blocking Ca(2+) and Ca(2+)-activated K(+) currents. We show, using intracellular recordings, that burstiness in low [Ca(2+)](o) persists even after truncating the apical dendrites, suggesting that bursts are generated by an interplay of membrane currents at or near the soma. To study the mechanisms of bursting, we have constructed a conductance-based, one-compartment model of CA1 pyramidal neurons. In this neuron model, reduced [Ca(2+)](o) is simulated by negatively shifting the activation curve of the persistent Na(+) current (I(NaP)) as indicated by recent experimental results. The neuron model accounts, with different parameter sets, for the diversity of firing patterns observed experimentally in both zero and normal [Ca(2+)](o). Increasing I(NaP) in the neuron model induces bursting and increases the number of spikes within a burst but is neither necessary nor sufficient for bursting. We show, using fast-slow analysis and bifurcation theory, that the M-type K(+) current (I(M)) allows bursting by shifting neuronal behavior between a silent and a tonically active state provided the kinetics of the spike generating currents are sufficiently, although not extremely, fast. We suggest that bursting in CA1 pyramidal cells can be explained by a single compartment "square bursting" mechanism with one slow variable, the activation of I(M).

Persistent Synchronized Bursting Activity in Cortical Tissues With Low Magnesium Concentration: A Modeling Study

We explore the mechanism of synchronized bursting activity with frequency of ∼10 Hz that appears in cortical tissues at low extracellular magnesium concentration [Mg2+]o. We hypothesize that this activity is persistent, namely coexists with the quiescent state and depends on slow N-methyl-d-aspartate (NMDA) conductances. To explore this hypothesis, we construct and investigate a conductance-based model of excitatory cortical networks. Population bursting activity can persist for physiological values of the NMDA decay time constant (∼100 ms). Neurons are synchronized at the time scale of bursts but not of single spikes. A reduced model of a cell coupled to itself can encompass most of this highly synchronized network behavior and is analyzed using the fast-slow method. Synchronized bursts appear for intermediate values of the NMDA conductance gNMDA if NMDA conductances are not too fast. Regular spiking activity appears for larger gNMDA. If the single cell is a conditional burster, persistent synchronized bursts become more robust. Weakly synchronized states appear for zero AMPA conductance gAMPA. Enhancing gAMPA increases both synchrony and the number of spikes within bursts and decreases the bursting frequency. Too strong gAMPA, however, prevents the activity because it enhances neuronal intrinsic adaptation. When [Mg2+]o is increased, higher gNMDA values are needed to maintain bursting activity. Bursting frequency decreases with [Mg2+]o, and the network is silent with physiological [Mg2+]o. Inhibition weakly decreases the bursting frequency if inhibitory cells receive enough NMDA-mediated excitation. This study explains the importance of conditional bursters in layer V in supporting epileptiform activity at low [Mg2+]o.

Background synaptic activity modulates the response of a modeled purkinje cell to paired afferent input

We studied the possible effects of background excitatory and inhibitory synaptic activity on Purkinje cell responses to paired pulse inputs using a morphologically and physiologically detailed compartmental model. Paired-pulse inputs were provided via synapses associated with the ascending segment of the granule cell axon in the presence of variable amounts of background synaptic input provided by parallel fibers and inhibitory interneurons. The results predict the amplitude and dynamics of the somatic and dendritic responses to paired ascending segment inputs are modulated by the level of background synaptic activity, the basal somatic firing rate of the Purkinje cell model does not indicate the type or degree of modulation, and the modulatory effects are mediated through dendritic calcium and calcium-activated potassium currents. These results have important consequences for the functional organization of cerebellar cortex as well as the more general issue of the modulatory as distinct from driving effects of synapses now also being considered in other regions of the mammalian brain including the cerebral cortex.

A minimal model for G protein-mediated synaptic facilitation and depression

G protein-coupled receptors are ubiquitous in neurons, as well as other cell types. Activation of receptors by hormones or neurotransmitters splits the G protein heterotrimer into Galpha and Gbetagamma subunits. It is now clear that Gbetagamma directly inhibits Ca2+ channels, putting them into a reluctant state. The effects of Gbetagamma depend on the specific beta and gamma subunits present, as well as the beta subunit isoform of the N-type Ca2+ channel. We describe a minimal mathematical model for the effects of G protein action on the dynamics of synaptic transmission. The model is calibrated by data obtained by transfecting G protein and Ca2+ channel subunits into tsA-201 cells. We demonstrate with numerical simulations that G protein action can provide a mechanism for either short-term synaptic facilitation or depression, depending on the manner in which G protein-coupled receptors are activated. The G protein action performs high-pass filtering of the presynaptic signal, with a filter cutoff that depends on the combination of G protein and Ca2+ channel subunits present. At stimulus frequencies above the cutoff, trains of single spikes are transmitted, while only doublets are transmitted at frequencies below the cutoff. Finally, we demonstrate that relief of G protein inhibition can contribute to paired-pulse facilitation.