Myongkeun Oh

Loss of phase-locking in non-weakly coupled inhibitory networks of type-I model neurons

Synchronization of excitable cells coupled by reciprocal inhibition is a topic of significant interest due to the important role that inhibitory synaptic interaction plays in the generation and regulation of coherent rhythmic activity in a variety of neural systems. While recent work revealed the synchronizing influence of inhibitory coupling on the dynamics of many networks, it is known that strong coupling can destabilize phase-locked firing. Here we examine the loss of synchrony caused by an increase in inhibitory coupling in networks of type-I Morris–Lecar model oscillators, which is characterized by a period-doubling cascade and leads to mode-locked states with alternation in the firing order of the two cells, as reported recently by Maran and Canavier (J Comput Nerosci, 2008) for a network of Wang-Buzsáki model neurons. Although alternating-order firing has been previously reported as a near-synchronous state, we show that the stable phase difference between the spikes of the two Morris–Lecar cells can constitute as much as 70% of the unperturbed oscillation period. Further, we examine the generality of this phenomenon for a class of type-I oscillators that are close to their excitation thresholds, and provide an intuitive geometric description of such “leap-frog” dynamics. In the Morris–Lecar model network, the alternation in the firing order arises under the condition of fast closing of K +  channels at hyperpolarized potentials, which leads to slow dynamics of membrane potential upon synaptic inhibition, allowing the presynaptic cell to advance past the postsynaptic cell in each cycle of the oscillation. Further, we show that non-zero synaptic decay time is crucial for the existence of leap-frog firing in networks of phase oscillators. However, we demonstrate that leap-frog spiking can also be obtained in pulse-coupled inhibitory networks of one-dimensional oscillators with a multi-branched phase domain, for instance in a network of quadratic integrate-and-fire model cells. Finally, for the case of a homogeneous network, we establish quantitative conditions on the phase resetting properties of each cell necessary for stable alternating-order spiking, complementing the analysis of Goel and Ermentrout (Physica D 163:191–216, 2002) of the order-preserving phase transition map.

Peptide Neuromodulation of Synaptic Dynamics in an Oscillatory Network

Although neuromodulation of synapses is extensively documented, its consequences in the context of network oscillations are not well known. We examine the modulation of synaptic strength and short-term dynamics in the crab pyloric network by the neuropeptide proctolin. Pyloric oscillations are driven by a pacemaker group which receives feedback through the inhibitory synapse from the lateral pyloric (LP) to pyloric dilator (PD) neurons. We show that proctolin modulates the spike-mediated and graded components of the LP to PD synapse. Proctolin enhances the graded component and unmasks a surprising heterogeneity in its dynamics where there is depression or facilitation depending on the amplitude of the voltage waveform of the presynaptic LP neuron. The spike-mediated component is influenced by the baseline membrane potential and is also enhanced by proctolin at all baseline potentials. In addition to direct modulation of this synapse, proctolin also changes the shape and amplitude of the presynaptic voltage waveform which additionally enhances synaptic output during ongoing activity. During ongoing oscillations, proctolin reduces the variability of cycle period but only when the LP to PD synapse is functionally intact. Using the dynamic clamp technique we find that the reduction in variability is a direct consequence of modulation of the LP to PD synapse. These results demonstrate that neuromodulation of synapses involves complex and interacting influences that target different synaptic components and dynamics as well as the presynaptic voltage waveform. At the network level, modulation of feedback inhibition can result in reduction of variability and enhancement of stable oscillatory output.

Neuromodulatory changes in short-term synaptic dynamics may be mediated by two distinct mechanisms of presynaptic calcium entry.

Although synaptic output is known to be modulated by changes in presynaptic calcium channels, additional pathways for calcium entry into the presynaptic terminal, such as non-selective channels, could contribute to modulation of short term synaptic dynamics. We address this issue using computational modeling. The neuropeptide proctolin modulates the inhibitory synapse from the lateral pyloric (LP) to the pyloric dilator (PD) neuron, two slow-wave bursting neurons in the pyloric network of the crab Cancer borealis. Proctolin enhances the strength of this synapse and also changes its dynamics. Whereas in control saline the synapse shows depression independent of the amplitude of the presynaptic LP signal, in proctolin, with high-amplitude presynaptic LP stimulation the synapse remains depressing while low-amplitude stimulation causes facilitation. We use simple calcium-dependent release models to explore two alternative mechanisms underlying these modulatory effects. In the first model, proctolin directly targets calcium channels by changing their activation kinetics which results in gradual accumulation of calcium with low-amplitude presynaptic stimulation, leading to facilitation. The second model uses the fact that proctolin is known to activate a non-specific cation current IMI. In this model, we assume that the MI channels have some permeability to calcium, modeled to be a result of slow conformation change after binding calcium. This generates a gradual increase in calcium influx into the presynaptic terminals through the modulatory channel similar to that described in the first model. Each of these models can explain the modulation of the synapse by proctolin but with different consequences for network activity.

Non-weak inhibition and phase resetting at negative values of phase in cells with fast-slow dynamics at hyperpolarized potentials

Phase response is a powerful concept in the analysis of both weakly and non-weakly perturbed oscillators such as regularly spiking neurons, and is applicable if the oscillator returns to its limit cycle trajectory between successive perturbations. When the latter condition is violated, a formal application of the phase return map may yield phase values outside of its definition domain; in particular, strong synaptic inhibition may result in negative values of phase. The effect of a second perturbation arriving close to the first one is undetermined in this case. However, here we show that for a Morris–Lecar model of a spiking cell with strong time scale separation, extending the phase response function definition domain to an additional negative value branch allows to retain the accuracy of the phase response approach in the face of such strong inhibitory coupling. We use the resulting extended phase response function to accurately describe the response of a Morris–Lecar oscillator to consecutive non-weak synaptic inputs. This method is particularly useful when analyzing the dynamics of three or more non-weakly coupled cells, whereby more than one synaptic perturbation arrives per oscillation cycle into each cell. The method of perturbation prediction based on the negative-phase extension of the phase response function may be applicable to other excitable cell models characterized by slow voltage dynamics at hyperpolarized potentials.

Synapses showing a preferred frequency in a reciprocally inhibitory neuronal network

Experimental and theoretical analysis suggest that a synapse capable of exhibiting both short-term facilitation and depression acts as a band-pass filter – where efficacy is maximal at an input frequency referred to as the preferred (resonance) frequency [3]. Different synapses from the same presynaptic cell can display different preferred frequencies [1,2]. It has been suggested that such frequency filtering by a synapse can provide an effective tool for selective communication between neurons [1]. However, little is known about the functional role of resonant synapses in network activity. Recent data from our lab indicates that inhibitory synapses in the crab pyloric central pattern generator (freq ~ 1 Hz) show resonance with a preferred presynaptic frequency of ~ 0.5 Hz (Fig. ​(Fig.1A).1A). Additionally, it is known that the bursting pacemaker neurons and some bursting follower neurons in this network have maximum impedance frequencies in response to periodic current injection (membrane resonance), all within the range of the network frequency [4]. To explore the potential role of synaptic preferred frequency, we use a model consisting of two neurons coupled with reciprocally inhibitory synapses that display preferred frequencies (Fig. ​(Fig.1B).1B). This model represents the coupling between the pyloric pacemaker neurons AB/PD and the follower neuron LP. The effect of preferred frequencies in our model is explored analytically as well as with simulations. In a network of two neurons coupled with resonance synapses, the frequency of each presynaptic neuron affects the strength of synapse and therefore the frequency of the postsynaptic neuron. Due to the recurrence of the network, these effects are reciprocal. We show that, regardless of cell type, the dynamics of each cell can be described by a logistic map which is composed with two functions: one characterizing the dependence of the synaptic conductance on the previous cycle period of presynaptic neuron, and the other determining the cycle period of postsynaptic neuron depending on this synaptic conductance. We show that the synapses with preferred frequencies can lead to complex network outputs, even in a simple two-cell network. In fact, the activity state of the two-cell network, as a function of the maximal synaptic strength (m) is characterized by a bifurcation diagram that involves a classical period-doubling cascade leading to chaotic dynamics (Fig. ​(Fig.1C).1C). The analytical prediction leading to the logistic-map bifurcation diagram can be reproduced in simulations, as we show using Morris-Lecar model neurons coupled with resonance synapses. Figure 1 A model of two reciprocally inhibitory neurons coupled with synapses showing a preferred frequency. A. Pyloric synapses show a preferred frequency as shown in voltage clamp recordings. B. Schematic of the model. C. Bifurcation diagram showing the period ...

Negative phase and extended spike-time response curve

The phase response curve (PRC) approach is a powerful tool in analyzing response of spiking cell to synaptic or other perturbations. The PRC-based analysis of the cell response involves describing the effect of perturbation as a change of the phase variable characterizing the state of the spiking cell, whereby the phase is always bounded on the interval [0, 1] or [0, 2π]. However, the extension of phase domain to negative values naturally arises when deriving phase return maps in the case of non-weak inhibition or larger networks coupled by three or more cells, as previously shown by Canavier and coworkers [1,2].

A mechanism underlying short-term synaptic dynamics regulated by neuromodulator based on kinetics of Ca currents

The crustacean stomatogastric nervous system (STNS) is one of the most extensively researched neural systems in studying the effects of neuromodulation. Previous studies have reported the actions of neuromodulators on intrinsic neuronal properties and synaptic strength in the STNS [2], but little is known about neuromodulatory effects on the short-term synaptic dynamics. We investigated the effect of the neuropeptide proctolin on the dynamics of the inhibitory synapse from the lateral pyloric (LP) to the pyloric dilator (PD) neuron in the crab pyloric network. Synaptic transmission between these neurons consists of spike-mediated and non-spike-mediated (graded) components. The graded component of this synapse shows short-term depression in control saline, but in the presence of proctolin, low-amplitude ( 30 mV) stimulation causes depression.

NEGATIVE PHASE AND LEADER SWITCHING IN NON-WEAKLY COUPLED TWO-CELL INHIBITORY NETWORKS

We examine the dynamics of a non-weakly coupled inhibitory network of two identical Morris-Lecar model neurons with type-I excitability, which was recently shown to exhibit stable alternating-order activity, whereby the spiking order of the two cells changes in each cycle of the oscillation. We provide an intuitive geometric description of such leader switching and demonstrate that the concept of negative phase allows to analyze the existence and stability of such alternating-order dynamics.