Convergent comodulation reduces interindividual variability of circuit output.

Ionic current levels of identified neurons vary substantially across individual animals. Yet, under similar conditions, neural circuit output can be remarkably similar, as evidenced in many motor systems. All neural circuits are influenced by multiple neuromodulators which provide flexibility to their output. These neuromodulators often overlap in their actions by modulating the same channel type or synapse, yet have neuron-specific actions resulting from distinct receptor expression. Because of this different receptor expression pattern, in the presence of multiple convergent neuromodulators, a common downstream target would be activated more uniformly in circuit neurons across individuals. We therefore propose that a baseline tonic (non-saturating) level of comodulation by convergent neuromodulators can reduce interindividual variability of circuit output. We tested this hypothesis in the pyloric circuit of the crab, Cancer borealis Multiple excitatory neuropeptides converge to activate the same voltage-gated current in this circuit, but different subsets of pyloric neurons have receptors for each peptide. We quantified the interindividual variability of the unmodulated pyloric circuit output by measuring the activity phases, cycle frequency and intraburst spike number and frequency. We then examined the variability in the presence of different combinations and concentrations of three neuropeptides. We found that at mid-level concentration (30 nM) but not at near-threshold (1 nM) or saturating (1 µM) concentrations, comodulation by multiple neuropeptides reduced the circuit output variability. Notably, the interindividual variability of response properties of an isolated neuron was not reduced by comodulation, suggesting that the reduction of output variability may emerge as a network effect.Significance Statement Neuromodulation has been explored as a mechanism to provide flexibility to the output of neural circuits. All neural circuits are subject to neuromodulation by multiple substances. These multiple neuromodulators often have convergent subcellular actions, and yet different circuit neurons express receptors for different neuromodulators. This pattern of cellular-level convergence and circuit-level divergence gives rise to the possibility that the presence of many modulators at sub-saturating concentrations may provide a consistent level of modulatory action of the circuit without qualitatively altering this output. This possibility indicates a different but complimentary role for neuromodulation: that convergent comodulation at sub-saturation levels reduces interindividual variability of neural circuit output.

The cerebellum directly modulates the substantia nigra dopaminergic activity.

Evidence of direct reciprocal connections between the cerebellum and basal ganglia has challenged the long-held notion that these structures function independently. While anatomical studies have suggested the presence of cerebellar projections to the substantia nigra pars compacta (SNc), the nature and function of these connections (Cb-SNc) is unknown. Here we show, in mice, that Cb-SNc projections form monosynaptic glutamatergic synapses with dopaminergic and non-dopaminergic neurons in the SNc. Optogenetic activation of Cb-SNc axons in the SNc is associated with increased SNc activity, elevated striatal dopamine levels and increased locomotion. During behavior, Cb-SNc projections are bilaterally activated before ambulation and unilateral lever manipulation. Cb-SNc projections show prominent activation for water reward and higher activation for sweet water, suggesting that the pathway also encodes reward value. Thus, the cerebellum directly, rapidly and effectively modulates basal ganglia dopamine levels and conveys information related to movement initiation, vigor and reward processing.

Modulation by Neuropeptides with Overlapping Targets Results in Functional Overlap in Oscillatory Circuit Activation.

Neuromodulation lends flexibility to neural circuit operation but the general notion that different neuromodulators sculpt neural circuit activity into distinct and characteristic patterns is complicated by interindividual variability. In addition, some neuromodulators converge onto the same signaling pathways, with similar effects on neurons and synapses. We compared the effects of three neuropeptides on the rhythmic pyloric circuit in the stomatogastric ganglion of male crabs, Cancer borealis Proctolin (PROC), crustacean cardioactive peptide (CCAP), and red pigment concentrating hormone (RPCH) activate the same modulatory inward current, I MI, and have convergent actions on synapses. However, while PROC targets all four neuron types in the core pyloric circuit, CCAP and RPCH target the same subset of only two neurons. After removal of spontaneous neuromodulator release, none of the neuropeptides restored the control cycle frequency, but all restored the relative timing between neuron types. Consequently, differences between neuropeptide effects were mainly found in the spiking activity of different neuron types. We performed statistical comparisons using the Euclidean distance in the multidimensional space of normalized output attributes to obtain a single measure of difference between modulatory states. Across preparations, the circuit output in PROC was distinguishable from CCAP and RPCH, but CCAP and RPCH were not distinguishable from each other. However, we argue that even between PROC and the other two neuropeptides, population data overlapped enough to prevent reliable identification of individual output patterns as characteristic for a specific neuropeptide. We confirmed this notion by showing that blind classifications by machine learning algorithms were only moderately successful.Significance Statement It is commonly assumed that distinct behaviors or circuit activities can be elicited by different neuromodulators. Yet it is unknown to what extent these characteristic actions remain distinct across individuals. We use a well-studied circuit model of neuromodulation to examine the effects of three neuropeptides, each known to produce a distinct activity pattern in controlled studies. We find that, when compared across individuals, the three peptides elicit activity patterns that are either statistically indistinguishable or show too much overlap to be labeled characteristic. We ascribe this to interindividual variability and overlapping subcellular actions of the modulators. Because both factors are common in all neural circuits, these findings have broad significance for understanding chemical neuromodulatory actions while considering interindividual variability.

Comodulation reduces interindividual variability of circuit output.

Ionic current levels of identified neurons vary substantially across individual animals. Yet, under similar conditions, neural circuit output can be remarkably similar, as evidenced in many motor systems. All neural circuits are influenced by multiple neuromodulators which provide flexibility to their output. These neuromodulators often overlap in their actions by modulating the same channel type or synapse, yet have neuron-specific actions resulting from distinct receptor expression. Because of this different receptor expression pattern, in the presence of multiple convergent neuromodulators, a common downstream target would be activated more uniformly in circuit neurons across individuals. We therefore propose that a baseline tonic (non-saturating) level of comodulation by convergent neuromodulators can reduce interindividual variability of circuit output. We tested this hypothesis in the pyloric circuit of the crab, Cancer borealis. Multiple excitatory neuropeptides converge to activate the same voltage-gated current in this circuit, but different subsets of pyloric neurons have receptors for each peptide. We quantified the interindividual variability of the unmodulated pyloric circuit output by measuring the activity phases, cycle frequency and intraburst spike number and frequency. We then examined the variability in the presence of different combinations and concentrations of three neuropeptides. We found that at mid-level concentration (30 nM) but not at near-threshold (1 nM) or saturating (1 µM) concentrations, comodulation by multiple neuropeptides reduced the circuit output variability. Notably, the interindividual variability of response properties of an isolated neuron was not reduced by comodulation, suggesting that the reduction of output variability may emerge as a network effect.

Modulation by neuropeptides with overlapping targets results in functional overlap in oscillatory circuit activation.

Neuromodulation lends flexibility to neural circuit operation but the general notion that different neuromodulators sculpt neural circuit activity into distinct and characteristic patterns is complicated by interindividual variability. In addition, some neuromodulators converge onto the same signaling pathways, with similar effects on neurons and synapses. We compared the effects of three neuropeptides on the rhythmic pyloric circuit in the crab Cancer borealis stomatogastric nervous system. Proctolin (PROC), crustacean cardioactive peptide (CCAP), and red pigment concentrating hormone (RPCH) all activate the same modulatory inward current, IMI, and have convergent actions on synapses. However, while PROC targets all four neuron types in the core pyloric circuit, CCAP and RPCH target the same subset of only two neurons. After removal of spontaneous neuromodulator release, none of the neuropeptides restored the control cycle frequency, but all restored the relative timing between neuron types. Consequently, differences between neuropeptide effects were mainly found in the spiking activity of different neuron types. We performed statistical comparisons using the Euclidean distance in the multidimensional space of normalized output attributes to obtain a single measure of difference between modulatory states. Across preparations, circuit output in PROC was distinguishable from CCAP and RPCH, but CCAP and RPCH were not distinguishable from each other. However, we argue that even between PROC and the other two neuropeptides, population data overlapped enough to prevent reliable identification of individual output patterns as characteristic for a specific neuropeptide. We confirmed this notion by showing that blind classifications by machine learning algorithms were only moderately successful.

On the Role of Theory and Modeling in Neuroscience.

In recent years, the field of neuroscience has gone through rapid experimental advances and a significant increase in the use of quantitative and computational methods. This growth has created a need for clearer analyses of the theory and modeling approaches used in the field. This issue is particularly complex in neuroscience because the field studies phenomena that cross a wide range of scales and often require consideration at varying degrees of abstraction, from precise biophysical interactions to the computations they implement. We argue that a pragmatic perspective of science, in which descriptive, mechanistic, and normative models and theories each play a distinct role in defining and bridging levels of abstraction, will facilitate neuroscientific practice. This analysis leads to methodological suggestions, including selecting a level of abstraction that is appropriate for a given problem, identifying transfer functions to connect models and data, and the use of models themselves as a form of experiment.

Distinct Mechanisms Underlie Electrical Coupling Resonance and Its Interaction with Membrane Potential Resonance.

Neurons in oscillatory networks often exhibit membrane potential resonance, a peak impedance at a non-zero input frequency. In electrically coupled oscillatory networks, the coupling coefficient (the ratio of post- and prejunctional voltage responses) could also show resonance. Such coupling resonance may emerge from the interaction between the coupling current and resonance properties of the coupled neurons, but this relationship has not been clearly described. Additionally, it is unknown if the gap-junction mediated electrical coupling conductance may have frequency dependence. We examined these questions by recording a pair of electrically coupled neurons in the oscillatory pyloric network of the crab Cancer borealis. We performed dual current- and voltage-clamp recordings and quantified the frequency preference of the coupled neurons, the coupling coefficient, the electrical conductance, and the postjunctional neuronal response. We found that all components exhibit frequency selectivity, but with distinct preferred frequencies. Mathematical and computational analysis showed that membrane potential resonance of the postjunctional neuron was sufficient to give rise to resonance properties of the coupling coefficient, but not the coupling conductance. A distinct coupling conductance resonance frequency therefore emerges either from other circuit components or from the gating properties of the gap junctions. Finally, to explore the functional effect of the resonance of the coupling conductance, we examined its role in synchronizing neuronal the activities of electrically coupled bursting model neurons. Together, our findings elucidate factors that produce electrical coupling resonance and the function of this resonance in oscillatory networks.

Cerebellar Contributions to the Basal Ganglia Influence Motor Coordination, Reward Processing, and Movement Vigor.

Both the cerebellum and the basal ganglia are known for their roles in motor control and motivated behavior. These two systems have been classically considered as independent structures that coordinate their contributions to behavior via separate cortico-thalamic loops. However, recent evidence demonstrates the presence of a rich set of direct connections between these two regions. Although there is strong evidence for connections in both directions, for brevity we limit our discussion to the better-characterized connections from the cerebellum to the basal ganglia. We review two sets of such connections: disynaptic projections through the thalamus and direct monosynaptic projections to the midbrain dopaminergic nuclei, the VTA and the SNc. In each case, we review the evidence for these pathways from anatomic tracing and physiological recordings, and discuss their potential functional roles. We present evidence that the disynaptic pathway through the thalamus is involved in motor coordination, and that its dysfunction contributes to motor deficits, such as dystonia. We then discuss how cerebellar projections to the VTA and SNc influence dopamine release in the respective targets of these nuclei: the NAc and the dorsal striatum. We argue that the cerebellar projections to the VTA may play a role in reward-based learning and therefore contribute to addictive behavior, whereas the projection to the SNc may contribute to movement vigor. Finally, we speculate how these projections may explain many of the observations that indicate a role for the cerebellum in mental disorders, such as schizophrenia.

Neuromodulation reduces interindividual variability of neuronal output.

In similar states, neural circuits produce similar outputs across individuals despite substantial interindividual variability in neuronal ionic conductances and synapses. Circuit states are largely shaped by neuromodulators that tune ionic conductances. It is therefore possible that, in addition to producing flexible circuit output, neuromodulators also contribute to output similarity despite varying ion channel expression. We studied whether neuromodulation at saturating concentrations can increase the output similarity of a single identified neuron across individual animals. Using the LP neuron of the crab stomatogastric ganglion (STG), we compared the variability of f-I curves and rebound properties in the presence of neuropeptides. The two neuropeptides we used converge to activate the same target current, which increases neuronal excitability. Output variability was lower in the presence of the neuropeptides, regardless of whether the neuropeptides significantly changed the mean of the corresponding parameter or not. However, the addition of the second neuropeptide did not add further to the reduction of variability. With a family of computational LP-like models, we explored how increased excitability and target variability contribute to output similarity and found two mechanisms: Saturation of the responses and a differential increase in baseline activity. Saturation alone can reduce the interindividual variability only if the population shares a similar ceiling for the responses. In contrast, reduction of variability due to the increase in baseline activity is independent of ceiling effects.Significance StatementThe activity of single neurons and neural circuits can be very similar across individuals even though the ionic currents underlying activity are variable. The mechanisms that compensate for the underlying variability and promote output similarity are poorly understood but may involve neuromodulation. Using an identified neuron, we show that neuropeptide modulation of excitability can reduce interindividual variability of response properties at a single-neuron level in two ways. First, the neuropeptide increases baseline excitability in a differential manner, resulting in similar response thresholds. Second, the neuropeptide increases excitability towards a shared saturation level, promoting similar maximal firing rates across individuals. Such tuning of neuronal excitability could be an important mechanism compensating for interindividual variability of ion channel expression.

Inter-Animal Variability in Activity Phase Is Constrained by Synaptic Dynamics in an Oscillatory Network.

The levels of voltage-gated and synaptic currents in the same neuron type can vary substantially across individuals. Yet, the phase relationships between neurons in oscillatory circuits are often maintained, even in the face of varying oscillation frequencies. We examined whether synaptic and intrinsic currents are matched to maintain constant activity phases across preparations, using the lateral pyloric (LP) neuron of the stomatogastric ganglion (STG) of the crab, Cancer borealis LP produces stable oscillatory bursts on release from inhibition, with an onset phase that is independent of oscillation frequency. We quantified the parameters that define the shape of the synaptic current inputs across preparations and found no linear correlations with voltage-gated currents. However, several synaptic parameters were correlated with oscillation period and burst onset phase, suggesting they may play a role in phase maintenance. We used dynamic clamp to apply artificial synaptic inputs and found that those synaptic parameters correlated with phase and period were ineffective in influencing burst onset. Instead, parameters that showed the least variability across preparations had the greatest influence. Thus, parameters that influence circuit phasing are constrained across individuals, while those that have little effect simply co-vary with phase and frequency.

Selective neural stimulation by leveraging electrophysiological differentiation and using pre-pulsing and non-rectangular waveforms.

Efforts on selective neural stimulation have concentrated on segregating axons based on their size and geometry. Nonetheless, axons of the white matter or peripheral nerves may also differ in their electrophysiological properties. The primary objective of this study was to investigate the possibility of selective activation of axons by leveraging an assumed level of diversity in passive (Cm & Gleak) and active membrane properties (Ktemp & Gnamax). First, the stimulus waveforms with hyperpolarizing (HPP) and depolarizing pre-pulsing (DPP) were tested on selectivity in a local membrane model. The default value of membrane capacitance (Cm) was found to play a critical role in sensitivity of the chronaxie time (Chr) and rheobase (Rhe) to variations of all the four membrane parameters. Decreasing the default value of Cm, and thus the passive time constant of the membrane, amplified the sensitivity to the active parameters, Ktemp and GNamax, on Chr. The HPP waveform could selectively activate neurons even if they were diversified by membrane leakage (Gleak) only, and produced higher selectivity than DPP when parameters are varied in pairs. Selectivity measures were larger when the passive parameters (Cm & Gleak) were varied together, compared to the active parameters. Second, this novel mechanism of selectivity was investigated with non-rectangular waveforms for the stimulating phase (and HPP) in the same local membrane model. Simulation results suggest that Kt2 is the most selective waveform followed by Linear and Gaussian waveforms. Traditional rectangular pulse was among the least selective of all. Finally, a compartmental axon model confirmed the main findings of the local model that Kt2 is the most selective, but rank ordered the other waveforms differently. These results suggest a potentially novel mechanism of stimulation selectivity, leveraging electrophysiological variations in membrane properties, that can lead to various neural prosthetic applications.

Mapping circuit dynamics during function and dysfunction.

Neural circuits can generate many spike patterns, but only some are functional. The study of how circuits generate and maintain functional dynamics is hindered by a poverty of description of circuit dynamics across functional and dysfunctional states. For example, although the regular oscillation of a central pattern generator is well characterized by its frequency and the phase relationships between its neurons, these metrics are ineffective descriptors of the irregular and aperiodic dynamics that circuits can generate under perturbation or in disease states. By recording the circuit dynamics of the well-studied pyloric circuit in Cancer borealis, we used statistical features of spike times from neurons in the circuit to visualize the spike patterns generated by this circuit under a variety of conditions. This approach captures both the variability of functional rhythms and the diversity of atypical dynamics in a single map. Clusters in the map identify qualitatively different spike patterns hinting at different dynamic states in the circuit. State probability and the statistics of the transitions between states varied with environmental perturbations, removal of descending neuromodulatory inputs, and the addition of exogenous neuromodulators. This analysis reveals strong mechanistically interpretable links between complex changes in the collective behavior of a neural circuit and specific experimental manipulations, and can constrain hypotheses of how circuits generate functional dynamics despite variability in circuit architecture and environmental perturbations.

Synaptic Dynamics Convey Differential Sensitivity to Input Pattern Changes in Two Muscles Innervated by the Same Motor Neurons.

Postsynaptic responses depend on input patterns as well as short-term synaptic plasticity, summation, and postsynaptic membrane properties, but the interactions of those dynamics with realistic input patterns are not well understood. We recorded the responses of the two pyloric dilator (PD) muscles, cpv2a and cpv2b, that are innervated by and receive identical periodic bursting input from the same two motor neurons in the lobster Homarus americanusCpv2a and cpv2b showed quantitative differences in membrane nonlinearities and synaptic summation. At a short timescale, responses in both muscles were dominated by facilitation, albeit with different frequency and time dependence. Realistic burst stimulations revealed more substantial differences. Across bursts, cpv2a showed transient depression, whereas cpv2b showed transient facilitation. Steady-state responses to bursting input also differed substantially. Neither muscle had a monotonic dependence on frequency, but cpv2b showed particularly pronounced bandpass filtering. Cpv2a was sensitive to changes in both burst frequency and intra-burst spike frequency, whereas, despite its much slower responses, cpv2b was largely insensitive to changes in burst frequency. Cpv2a was sensitive to both burst duration and number of spikes per burst, whereas cpv2b was sensitive only to the former parameter. Neither muscle showed consistent sensitivity to changes in the overall spike interval structure, but cpv2b was surprisingly sensitive to changes in the first intervals in each burst, a parameter known to be regulated by dopamine (DA) modulation of spike propagation of the presynaptic axon. These findings highlight how seemingly minor circuit output changes mediated by neuromodulation could be read out differentially at the two synapses.

Frequency-Dependent Action of Neuromodulation.

In oscillatory circuits, some actions of neuromodulators depend on the oscillation frequency. However, the mechanisms are poorly understood. We explored this problem by characterizing neuromodulation of the lateral pyloric (LP) neuron of the crab stomatogastric ganglion (STG). Many peptide modulators, including proctolin, activate the same ionic current (IMI) in STG neurons. Because IMI is fast and non-inactivating, its peak level does not depend on the temporal properties of neuronal activity. We found, however, that the amplitude and peak time of the proctolin-activated current in LP is frequency dependent. Because frequency affects the rate of voltage change, we measured these currents with voltage ramps of different slopes and found that proctolin activated two kinetically distinct ionic currents: the known IMI, whose amplitude is independent of ramp slope or direction, and an inactivating current (IMI-T), which was only activated by positive ramps and whose amplitude increased with increasing ramp slope. Using a conductance-based model we found that IMI and IMI-T make distinct contributions to the bursting activity, with IMI increasing the excitability, and IMI-T regulating the burst onset by modifying the postinhibitory rebound in a frequency-dependent manner. The voltage dependence and partial calcium permeability of IMI-T is similar to other characterized neuromodulator-activated currents in this system, suggesting that these are isoforms of the same channel. Our computational model suggests that calcium permeability may allow this current to also activate the large calcium-dependent potassium current in LP, providing an additional mechanism to regulate burst termination. These results demonstrate a mechanism for frequency-dependent actions of neuromodulators.

Mutual Suppression of Proximal and Distal Axonal Spike Initiation Determines the Output Patterns of a Motor Neuron.

Axonal spike initiation at sites far from somatodendritic integration occurs in a range of systems, but its contribution to neuronal output activity is not well understood. We studied the interactions of distal and proximal spike initiation in an unmyelinated motor axon of the stomatogastric nervous system in the lobster, Homarus americanus. The peripheral axons of the pyloric dilator (PD) neurons generate tonic spiking in response to dopamine application. Centrally generated bursting activity and peripheral spike initiation had mutually suppressive effects. The two PD neurons and the electrically coupled oscillatory anterior burster (AB) neuron form the pacemaker ensemble of the pyloric central pattern generator, and antidromic invasion of central compartments by peripherally generated spikes caused spikelets in AB. Antidromic spikes suppressed burst generation in an activity-dependent manner: slower rhythms were diminished or completely disrupted, while fast rhythmic activity remained robust. Suppression of bursting was based on interference with the underlying slow wave oscillations in AB and PD, rather than a direct effect on spike initiation. A simplified multi-compartment circuit model of the pacemaker ensemble replicated this behavior. Antidromic activity disrupted slow wave oscillations by resetting the inward and outward current trajectories in each spike interval. Centrally generated bursting activity in turn suppressed peripheral spike initiation in an activity-dependent manner. Fast bursting eliminated peripheral spike initiation, while slower bursting allowed peripheral spike initiation to continue during the intervals between bursts. The suppression of peripheral spike initiation was associated with a small after-hyperpolarization in the sub-millivolt range. A realistic model of the PD axon replicated this behavior and showed that a sub-millivolt cumulative after-hyperpolarization across bursts was sufficient to eliminate peripheral spike initiation. This effect was based on the dynamic interaction between slow activity-dependent hyperpolarization caused by the Na+/K+-pump and inward rectification through the hyperpolarization-activated inward current, I h . These results demonstrate that interactions between different spike initiation sites based on spike propagation can shift the relative contributions of different types of activity in an activity-dependent manner. Therefore, distal axonal spike initiation can play an important role in shaping neural output, conditional on the relative level of centrally generated activity.

The differential contribution of pacemaker neurons to synaptic transmission in the pyloric network of the Jonah crab, Cancer borealis.

Many neurons receive synchronous input from heterogeneous presynaptic neurons with distinct properties. An instructive example is the crustacean stomatogastric pyloric circuit pacemaker group, consisting of the anterior burster (AB) and pyloric dilator (PD) neurons, which are active synchronously and exert a combined synaptic action on most pyloric follower neurons. Previous studies in lobster have indicated that AB is glutamatergic, whereas PD is cholinergic. However, although the stomatogastric system of the crab Cancer borealis has become a preferred system for exploration of cellular and synaptic basis of circuit dynamics, the pacemaker synaptic output has not been carefully analyzed in this species. We examined the synaptic properties of these neurons using a combination of single-cell mRNA analysis, electrophysiology, and pharmacology. The crab PD neuron expresses high levels of choline acetyltransferase and the vesicular acetylcholine transporter mRNAs, hallmarks of cholinergic neurons. In contrast, the AB neuron expresses neither cholinergic marker but expresses high levels of vesicular glutamate transporter mRNA, consistent with a glutamatergic phenotype. Notably, in the combined synapses to follower neurons, 70-75% of the total current was blocked by putative glutamatergic blockers, but short-term synaptic plasticity remained unchanged, and although the total pacemaker current in two follower neuron types was different, this difference did not contribute to the phasing of the follower neurons. These findings provide a guide for similar explorations of heterogeneous synaptic connections in other systems and a baseline in this system for the exploration of the differential influence of neuromodulators.NEW & NOTEWORTHY The pacemaker-driven pyloric circuit of the Jonah crab stomatogastric nervous system is a well-studied model system for exploring circuit dynamics and neuromodulation, yet the understanding of the synaptic properties of the two pacemaker neuron types is based on older analyses in other species. We use single-cell PCR and electrophysiology to explore the neurotransmitters used by the pacemaker neurons and their distinct contribution to the combined synaptic potentials.

Frequency-dependent responses of neuronal models to oscillatory inputs in current versus voltage clamp.

Action potential generation in neurons depends on a membrane potential threshold and therefore on how subthreshold inputs influence this voltage. In oscillatory networks, for example, many neuron types have been shown to produce membrane potential ([Formula: see text]) resonance: a maximum subthreshold response to oscillatory inputs at a nonzero frequency. Resonance is usually measured by recording [Formula: see text] in response to a sinusoidal current ([Formula: see text]), applied at different frequencies (f), an experimental setting known as current clamp (I-clamp). Several recent studies, however, use the voltage clamp (V-clamp) method to control [Formula: see text] with a sinusoidal input at different frequencies [[Formula: see text]] and measure the total membrane current ([Formula: see text]). The two methods obey systems of differential equations of different dimensionality, and while I-clamp provides a measure of electrical impedance [[Formula: see text]], V-clamp measures admittance [[Formula: see text]]. We analyze the relationship between these two measurement techniques. We show that, despite different dimensionality, in linear systems the two measures are equivalent: [Formula: see text]. However, nonlinear model neurons produce different values for Z and [Formula: see text]. In particular, nonlinearities in the voltage equation produce a much larger difference between these two quantities than those in equations of recovery variables that describe activation and inactivation kinetics. Neurons are inherently nonlinear, and notably, with ionic currents that amplify resonance, the voltage clamp technique severely underestimates the current clamp response. We demonstrate this difference experimentally using the PD neurons in the crab stomatogastric ganglion. These findings are instructive for researchers who explore cellular mechanisms of neuronal oscillations.

Short-term synaptic dynamics control the activity phase of neurons in an oscillatory network.

In oscillatory systems, neuronal activity phase is often independent of network frequency. Such phase maintenance requires adjustment of synaptic input with network frequency, a relationship that we explored using the crab, Cancer borealis, pyloric network. The burst phase of pyloric neurons is relatively constant despite a > two fold variation in network frequency. We used noise input to characterize how input shape influences burst delay of a pyloric neuron, and then used dynamic clamp to examine how burst phase depends on the period, amplitude, duration, and shape of rhythmic synaptic input. Phase constancy across a range of periods required a proportional increase of synaptic duration with period. However, phase maintenance was also promoted by an increase of amplitude and peak phase of synaptic input with period. Mathematical analysis shows how short-term synaptic plasticity can coordinately change amplitude and peak phase to maximize the range of periods over which phase constancy is achieved.

Distinct Co-Modulation Rules of Synapses and Voltage-Gated Currents Coordinate Interactions of Multiple Neuromodulators.

Multiple neuromodulators act in concert to shape the properties of neural circuits. Different neuromodulators usually activate distinct receptors but can have overlapping targets. Therefore, circuit output depends on neuromodulator interactions at shared targets, a poorly understood process. We explored quantitative rules of co-modulation of two principal targets of neuromodulation: synapses and voltage-gated ionic currents. In the stomatogastric ganglion of the male crab Cancer borealis, the neuropeptides proctolin (Proc) and the crustacean cardioactive peptide (CCAP) modulate synapses of the pyloric circuit and activate a voltage-gated current (IMI) in multiple neurons. We examined the validity of a simple dose-dependent quantitative rule, that co-modulation by Proc and CCAP is predicted by the linear sum of the individual effects of each modulator up to saturation. We found that this rule is valid for co-modulation of synapses, but not for the activation of IMI, in which co-modulation was sublinear. The predictions for the co-modulation of IMI activation were greatly improved if we assumed that the intracellular pathways activated by two peptide receptors inhibit one another. These findings suggest that the pathways activated by two neuromodulators could have distinct interactions, leading to distinct co-modulation rules for different targets even in the same neuron. Given the evolutionary conservation of neuromodulator receptors and signaling pathways, such distinct rules for co-modulation of different targets are likely to be common across neuronal circuits.SIGNIFICANCE STATEMENT We examine the quantitative rules of co-modulation at multiple shared targets, the first such characterization to our knowledge. Our results show that dose-dependent co-modulation of distinct targets in the same cells by the same two neuromodulators follows different rules: co-modulation of synaptic currents is linearly additive up to saturation, whereas co-modulation of the voltage-gated ionic current targeted in a single neuron is nonlinear, a mechanism that is likely generalizable. Given that all neural systems are multiply modulated and neuromodulators often act on shared targets, these findings and the methodology could guide studies to examine dynamic actions of neuromodulators at the biophysical and systems level in sensory and motor functions, sleep/wake regulation, and cognition.

Phase-locking and bistability in neuronal networks with synaptic depression.

We consider a recurrent network of two oscillatory neurons that are coupled with inhibitory synapses. We use the phase response curves of the neurons and the properties of short-term synaptic depression to define Poincaré maps for the activity of the network. The fixed points of these maps correspond to phase-locked modes of the network. Using these maps, we analyze the conditions that allow short-term synaptic depression to lead to the existence of bistable phase-locked, periodic solutions. We show that bistability arises when either the phase response curve of the neuron or the short-term depression profile changes steeply enough. The results apply to any Type I oscillator and we illustrate our findings using the Quadratic Integrate-and-Fire and Morris-Lecar neuron models.