Shape Of Phase Response Curve Research Articles

Phase response theory explains cluster formation in sparsely but strongly connected inhibitory neural networks and effects of jitter due to sparse connectivity.

We show how to predict whether a neural network will exhibit global synchrony (a one-cluster state) or a two-cluster state based on the assumption of pulsatile coupling and critically dependent upon the phase response curve (PRC) generated by the appropriate perturbation from a partner cluster. Our results hold for a monotonically increasing (meaning longer delays as the phase increases) PRC, which likely characterizes inhibitory fast-spiking basket and cortical low-threshold-spiking interneurons in response to strong inhibition. Conduction delays stabilize synchrony for this PRC shape, whereas they destroy two-cluster states, the former by avoiding a destabilizing discontinuity and the latter by approaching it. With conduction delays, stronger coupling strength can promote a one-cluster state, so the weak coupling limit is not applicable here. We show how jitter can destabilize global synchrony but not a two-cluster state. Local stability of global synchrony in an all-to-all network does not guarantee that global synchrony can be observed in an appropriately scaled sparsely connected network; the basin of attraction can be inferred from the PRC and must be sufficiently large. Two-cluster synchrony is not obviously different from one-cluster synchrony in the presence of noise and may be the actual substrate for oscillations observed in the local field potential (LFP) and the electroencephalogram (EEG) in situations where global synchrony is not possible. Transitions between cluster states may change the frequency of the rhythms observed in the LFP or EEG. Transitions between cluster states within an inhibitory subnetwork may allow more effective recruitment of pyramidal neurons into the network rhythm. NEW & NOTEWORTHY We show that jitter induced by sparse connectivity can destabilize global synchrony but not a two-cluster state with two smaller clusters firing alternately. On the other hand, conduction delays stabilize synchrony and destroy two-cluster states. These results hold if each cluster exhibits a phase response curve similar to one that characterizes fast-spiking basket and cortical low-threshold-spiking cells for strong inhibition. Either a two-cluster or a one-cluster state might provide the oscillatory substrate for neural computations.

Effect of Phase Response Curve Shape and Synaptic Driving Force on Synchronization of Coupled Neuronal Oscillators.

The role of the phase response curve (PRC) shape on the synchrony of synaptically coupled oscillating neurons is examined. If the PRC is independent of the phase, because of the synaptic form of the coupling, synchrony is found to be stable for both excitatory and inhibitory coupling at all rates, whereas the antisynchrony becomes stable at low rates. A faster synaptic rise helps extend the stability of antisynchrony to higher rates. If the PRC is not constant but has a profile like that of a leaky integrate-and-fire model, then, in contrast to the earlier reports that did not include the voltage effects, mutual excitation could lead to stable synchrony provided the synaptic reversal potential is below the voltage level the neuron would have reached in the absence of the interaction and threshold reset. This level is controlled by the applied current and the leakage parameters. Such synchrony is contingent on significant phase response (that would result, for example, by a sharp PRC jump) occurring during the synaptic rising phase. The rising phase, however, does not contribute significantly if it occurs before the voltage spike reaches its peak. Then a stable near-synchronous state can still exist between type 1 PRC neurons if the PRC shows a left skewness in its shape. These results are examined comprehensively using perfect integrate-and-fire, leaky integrate-and-fire, and skewed PRC shapes under the assumption of the weakly coupled oscillator theory applied to synaptically coupled neuron models.

Are phase resetting curves tunable?

Neurons are excitable cells capable of sustaining high amplitude membrane potential oscillations called action potential (AP). There are two classes of repetitively firing neurons: Class 1 characterized by a continuously and arbitrarily low tunable firing frequency, or Class 2 characterized by a minimum firing frequency threshold [1]. The ionic channels and the morphology of the cells determine the characteristics of repetitive firing, such as the intrinsic period Pi, the firing threshold, or the duration of the AP. For the study of phase locked modes and synchrony of neural populations a phenomenological understanding of neural activity suffices. In such cases, the phase response curve (PRC) rather than a detailed biophysical model of the neuron provides accurate and quick results. A PRC tabulates the transient changes in the firing period of a neuron due to an external perturbation, usually a brief current or synaptic conductance pulse [2,3]. The most significant effect of a perturbation occurs during the cycle that contains the perturbation and is quantified by the first order PRC, i.e. F1(φ) = P1/Pi - 1, where P1 is the transiently modified duration of the current cycle due to a perturbation applied at the stimulus time ts or phase φ = ts/Pi (Figure ​(Figure1A).1A). There are two types of PRCs: Type II PRC that has an almost sinusoidal shape and comparable sizes of the positive and negative lobes, and Type I PRC that has a disproportionate ratio of the two lobes (see Figure ​Figure1B).1B). When Type I neurons are part of neural network they can predominantly speed up (or only slow down) their rhythm, whereas Type II neurons synchronize more readily due to their ability to both slow down and speed up. While it is clear that every PRCs are bimodal, the purpose of this study is to investigate what parameters determine their shape. We found that the phase space bifurcation structure determines not only the PRC’s type but also how easily the shape of the PRC can be tuned such that a “typical” Type I PRC can be smoothly converted into a Type II PRC. We found that nonlinear oscillators characterized both by saddle-node of invariant circle (SNIC) and Andronov-Hopf (HB) bifurcations, such as Morris-Lecar (ML) model neuron, the PRC can be smoothly adjusted from a pure Type I to a Type II. For example, by slowly increasing a bias current, ML model reaches a SNIC bifurcation and produces a typical Type I PRC approximately captured by the generic infinitesimal PRC iPRC1(φ) = 1 - cos(φ). Further increasing the bias current brings the model closer to a HB that produces a typical Type II infinitesimal PRC iPRC2(φ) = sin(φ + α), with α a constant phase. For any intermediate value of the bias current the actual shape of the infinitesimal PRC is a linear combination of iPRC1 and iPRC2. Figure 1 (A) Intrinsic oscillations (continuous line) and transiently perturbed period (dashed line). (B) A Type I PRC is bimodal with one large lobe of amplitude M and the other with a disproportionately smaller amplitude m.

Effect of Phase Response Curve Skewness on Synchronization of Electrically Coupled Neuronal Oscillators

We investigate why electrically coupled neuronal oscillators synchronize or fail to synchronize using the theory of weakly coupled oscillators. Stability of synchrony and antisynchrony is predicted analytically and verified using numerical bifurcation diagrams. The shape of the phase response curve (PRC), the shape of the voltage time course, and the frequency of spiking are freely varied to map out regions of parameter spaces that hold stable solutions. We find that type 1 and type 2 PRCs can hold both synchronous and antisynchronous solutions, but the shape of the PRC and the voltage determine the extent of their stability. This is achieved by introducing a five-piecewise linear model to the PRC and a three-piecewise linear model to the voltage time course, and then analyzing the resultant eigenvalue equations that determine the stability of the phase-locked solutions. A single time parameter defines the skewness of the PRC, and another single time parameter defines the spike width and frequency. Our approach gives a comprehensive picture of the relation of the PRC shape, voltage time course, and stability of the resultant synchronous and antisynchronous solutions.

Effect of phase response curve skew on synchronization with and without conduction delays

A central problem in cortical processing including sensory binding and attentional gating is how neurons can synchronize their responses with zero or near-zero time lag. For a spontaneously firing neuron, an input from another neuron can delay or advance the next spike by different amounts depending upon the timing of the input relative to the previous spike. This information constitutes the phase response curve (PRC). We present a simple graphical method for determining the effect of PRC shape on synchronization tendencies and illustrate it using type 1 PRCs, which consist entirely of advances (delays) in response to excitation (inhibition). We obtained the following generic solutions for type 1 PRCs, which include the pulse-coupled leaky integrate and fire model. For pairs with mutual excitation, exact synchrony can be stable for strong coupling because of the stabilizing effect of the causal limit region of the PRC in which an input triggers a spike immediately upon arrival. However, synchrony is unstable for short delays, because delayed inputs arrive during a refractory period and cannot trigger an immediate spike. Right skew destabilizes antiphase and enables modes with time lags that grow as the conduction delay is increased. Therefore, right skew favors near synchrony at short conduction delays and a gradual transition between synchrony and antiphase for pairs coupled by mutual excitation. For pairs with mutual inhibition, zero time lag synchrony is stable for conduction delays ranging from zero to a substantial fraction of the period for pairs. However, for right skew there is a preferred antiphase mode at short delays. In contrast to mutual excitation, left skew destabilizes antiphase for mutual inhibition so that synchrony dominates at short delays as well. These pairwise synchronization tendencies constrain the synchronization properties of neurons embedded in larger networks.

Influences of membrane properties on phase response curve and synchronization stability in a model globus pallidus neuron

The activity patterns of the globus pallidus (GPe) and subthalamic nucleus (STN) are closely associated with motor function and dysfunction in the basal ganglia. In the pathological state caused by dopamine depletion, the STN-GPe network exhibits rhythmic synchronous activity accompanied by rebound bursts in the STN. Therefore, the mechanism of activity transition is a key to understand basal ganglia functions. As synchronization in GPe neurons could induce pathological STN rebound bursts, it is important to study how synchrony is generated in the GPe. To clarify this issue, we applied the phase-reduction technique to a conductance-based GPe neuronal model in order to derive the phase response curve (PRC) and interaction function between coupled GPe neurons. Using the PRC and interaction function, we studied how the steady-state activity of the GPe network depends on intrinsic membrane properties, varying ionic conductances on the membrane. We noted that a change in persistent sodium current, fast delayed rectifier Kv3 potassium current, M-type potassium current and small conductance calcium-dependent potassium current influenced the PRC shape and the steady state. The effect of those currents on the PRC shape could be attributed to extension of the firing period and reduction of the phase response immediately after an action potential. In particular, the slow potassium current arising from the M-type potassium and the SK current was responsible for the reduction of the phase response. These results suggest that the membrane property modulation controls synchronization/asynchronization in the GPe and the pathological pattern of STN-GPe activity.

The influence of stationary synaptic activity on the PRC

A useful measurable property of a neural oscillator is its Phase Response Curve (PRC). PRCs measure the phase-shift resulting from perturbing the oscillator with a brief stimulus at different times of the cycle. They have been extensively used to understand the synchronous activity patterns emerging from a network of weakly coupled oscillators. PRCs have been classified into two types: type I (PRC is always positive) and type II (PRC has positive and negative regions) [1]. Theoretical results [2] have shown that the type of PRC combined with the temporal dynamics of the synapses yield different synchronization properties when two neurons are coupled together (neurons can synchronize in-phase, out of phase or in anti-phase). PRCs are typically measured in vitro, considering only the intrinsic properties of the neuron. However, in vivo neurons constantly receive background synaptic inputs that play an important role sculpting the dynamics of neurons. Indeed experimental data showed that membrane excitability [3] can change in response to variations in background synaptic activity [4]. In this work we study the effects of the background synaptic activity on the shape of the Phase Response Curve, and its synchronization properties. To perform this study, we consider two neuron models: the Wang-Buzsaki model [5] and the Morris-Lecar model [6]. We explore the effect of a constant excitatory and inhibitory synaptic conductance input (that can be seen as an average of the background input) on the type of membrane excitability and PRC shape in the spiking regime. We found that changes in the mean background conductances in a biologically plausible range [7] lead to changes in the type of PRC. As we increased the inhibitory conductance, for a constant value of the excitatory one, we observed a switch from type I to type II PRC. We correlated the shape of the PRC with the synchronization properties. We studied the effect of the temporal dynamics of synaptic activation on the synchronization properties of a coupled pair of neurons, as we switched them from type I to type II PRC. We characterized how solutions change with these parameters in a network motif of two reciprocally coupled neurons.

Phase response analysis during in vivo-like high conductance states; dendritic SK determines the mean and variance of responses to dendritic excitation

A neuron’s phase response curve (PRC) describes how synaptic inputs at different times during the spike cycle affect the timing of subsequent spikes, and PRC analysis is a powerful technique for predicting and interpreting the emergence of synchronous modes in synaptically coupled networks and neuronal populations receiving common input. However, neuronal PRCs are typically measured during intrinsic pacemaking which may not reflect neuronal excitability and dynamics during high conductance states generated by complex network activity in vivo. Using a full morphological model of a globus pallidus (GP) neuron we have recently demonstrated that during intrinsic pacemaking, somatic PRCs for GP neurons are type I, i.e. excitatory inputs at all phases of the spike cycle advance the spontaneous spiking rhythm1. We also demonstrated that synaptic excitation of the distal dendrite can paradoxically delay subsequent spiking when delivered at some phases of the spike cycle (yielding a type II PRC) as a consequence of dendritic activation of the small conductance calcium-activated potassium current, SK1. Since during high conductance states spike timing is determined by a balance between intrinsic mechanisms and synaptic input fluctuations, in this study we investigated how somatic and dendritic phase response properties of the GP model are affected by ongoing stochastic synaptic background activity. We generated high conductance states in the model by applying synaptic backgrounds composed of randomly-timed excitatory and inhibitory synaptic inputs at 1022 GABA synapses and 100 AMPA synapses distributed throughout the dendrite. By varying the synaptic gain and input frequency parameters across the physiological range for inputs to GP, we achieved a diverse set of high conductance states characterized by sub-threshold voltage fluctuations, irregular spiking, and elevated membrane conductance (Figure ​(Figure1A1A &1B) and spanning the range of spike frequencies observed in vivo (15-45 Hz). For each synaptic background parameter set, we generated 100 single-trial PRCs by delivering a single 2.5 nS AMPA-synaptic input to either the soma or distal dendrite at each of 72 time-points (in separate simulations) within the first spike cycle of each of 100 control spike trains (Figure ​(Figure1C1C &1D). We then averaged the single-trial PRCs for each synaptic background to evaluate the dependence of PRC shape on stimulus location, spike frequency, and dendritic SK conductance, in addition to the gain and input frequencies of synaptic backgrounds. Next, we analyzed on a trial-by-trial basis the interactions of PRC stimuli with transient fluctuations in the synaptic background leading to added or skipped spikes. We determined that dendritic SK underlies both the incidence of skipped spikes and the dependence of skipped spike events on stimulus phase. Interestingly, the input-phase dependence of skipped spike events mirrored the phase-dependent variance in the PRC itself (which we plotted as phase response-variance curves, PRVCs). This indicates that dendritic SK determines not only the mean response, but also the variance of responses, to excitatory dendritic inputs. Average somatic and dendritic PRCs were type I and type II, respectively, and are likely to represent the average behavior of populations of GP neurons in response to shared excitation. Furthermore, in-as-much as the GP model and simulated synaptic backgrounds are physiologically realistic, the variance in the model’s responses across trials directly reflects a major source of spike time variance across populations of GP neurons in vivo. Figure 1 A&B. Total synaptic conductance (top) and current (bottom) for synaptic backgrounds composed of 1 nS (A) or 2 nS (B) unitary inputs. C&D. Perturbations of control spike trains (black) by somatic (C) or dendritic (D2D)(D) stimuli. (Trace ...

A qualitative optimization technique for biophysical neuron models with many parameters

We present a novel computational technique that enables more efficient optimization of qualitative features in biophysical neural models. We extend the idea of multiple objective optimization for a single comparison modality (such as comparison of individual voltage traces) [1] into multiple modalities. For instance, features may be defined in terms of measurements from different experimental scenarios, and might include not only action potential spike shape characteristics (Cf. [1,2]) but also complex features such as the frequency response curve, and phase response curve (PRC). Multiple fuzzy feature modalities provide additional ways to distinguish solutions that might otherwise appear similarly fit in the restrictive view of a single modality and with crisp measures of fitness. An ill-posed optimization problem with many local minima typically arises from relative insensitivities of some parameters to features in a single modality, or parameter co-variation/modulation [2]. We use an implementation in the PyDSTool software [3] to demonstrate how the multi-modal approach can be prepared and embedded in conventional gradient-based or global optimization algorithms. Among other applications, we discuss the transformation over 25 model parameters of a Hodgkin-Huxley neuron model with a Type I PRC [4] to a Type II PRC, using a goal PRC shape and other periodic trajectory statistics as fuzzy objective modalities.

Cholinergic Neuromodulation Changes Phase Response Curve Shape and Type in Cortical Pyramidal Neurons

Spike generation in cortical neurons depends on the interplay between diverse intrinsic conductances. The phase response curve (PRC) is a measure of the spike time shift caused by perturbations of the membrane potential as a function of the phase of the spike cycle of a neuron. Near the rheobase, purely positive (type I) phase-response curves are associated with an onset of repetitive firing through a saddle-node bifurcation, whereas biphasic (type II) phase-response curves point towards a transition based on a Hopf-Andronov bifurcation. In recordings from layer 2/3 pyramidal neurons in cortical slices, cholinergic action, consistent with down-regulation of slow voltage-dependent potassium currents such as the M-current, switched the PRC from type II to type I. This is the first report showing that cholinergic neuromodulation may cause a qualitative switch in the PRCs type implying a change in the fundamental dynamical mechanism of spike generation.

Modelling of circadian rhythms in Drosophila incorporating the interlocked PER/TIM and VRI/PDP1 feedback loops

Circadian rhythms of gene activity, metabolism, physiology and behaviour are observed in all the eukaryotes and some prokaryotes. In this study, we present a model to represent the transcriptional regulatory network essential for the circadian rhythmicity in Drosophila. The model incorporates the transcriptional feedback loops revealed so far in the network of the circadian clock (PER/TIM and VRI/PDP1 loops). Conventional Hill functions are not assumed to describe the regulation of genes, instead of the explicit reactions of binding and unbinding processes of transcription factors to promoters are modelled. The model simulates sustained circadian oscillations in mRNA and protein concentrations in constant darkness in agreement with experimental observations. It also simulates entrainment by light-dark cycles, disappearance of the rhythmicity in constant light and the shape of phase response curves resembling that of the experimental results. The model is robust over a wide range of parameter variations. In addition, the simulated E-box mutation, per S and per L mutants are similar to that observed in the experiments. The deficiency between the simulated mRNA levels and experimental observations in per 01 , tim 01 and clk Jrk mutants suggests some difference on the part of the model from reality.

Diazepam phase shifts the circadian clock of the field mouseMus booduga

Single 2h administration of diazepam (benzodiazepine) in 3.5% ethanol solution was found to evoke advance and delay phase shifts in the locomotor activity rhythm in the field mouseMus booduga. Through such pulsed administration of diazepam at various phases of circadian rhythm a phase response curve could be constructed. Phase advance occurred during early subjective day (CT 2) and phase delays were observed in the remaining phases. The shape of the diazepam phase response curve is similar to the general shape of the phase response curves generated by intraperitoneal injections of other benzodiazepines in hamsters. The phase shifting action of diazepam may be explained by its agonistic action on the neurotransmitter gamma-aminobutyric acid.