Modeling circadian oscillations with interlocking positive and negative feedback loops.

Both positive and negative feedback loops of transcriptional regulation have been proposed to be important for the generation of circadian rhythms. To test the sufficiency of the proposed mechanisms, two differential equation-based models were constructed to describe the Neurospora crassa and Drosophila melanogaster circadian oscillators. In the model of the Neurospora oscillator, FRQ suppresses frq transcription by binding to a complex of the transcriptional activators WC-1 and WC-2, thus yielding negative feedback. FRQ also activates synthesis of WC-1, which in turn activates frq transcription, yielding positive feedback. In the model of the Drosophila oscillator, PER and TIM are represented by a "lumped" variable, "PER." PER suppresses its own transcription by binding to the transcriptional regulator dCLOCK, thus yielding negative feedback. PER also binds to dCLOCK to de-repress dclock, and dCLOCK in turn activates per transcription, yielding positive feedback. Both models displayed circadian oscillations that were robust to parameter variations and to noise and that entrained to simulated light/dark cycles. Circadian oscillations were only obtained if time delays were included to represent processes not modeled in detail (e.g., transcription and translation). In both models, oscillations were preserved when positive feedback was removed.

Modeling transcriptional control in gene networks--methods, recent results, and future directions.

Mathematical models are useful for providing a framework for integrating data and gaining insights into the static and dynamic behavior of complex biological systems such as networks of interacting genes. We review the dynamic behaviors expected from model gene networks incorporating common biochemical motifs, and we compare current methods for modeling genetic networks. A common modeling technique, based on simply modeling genes as ON-OFF switches, is readily implemented and allows rapid numerical simulations. However, this method may predict dynamic solutions that do not correspond to those seen when systems are modeled with a more detailed method using ordinary differential equations. Until now, the majority of gene network modeling studies have focused on determining the types of dynamics that can be generated by common biochemical motifs such as feedback loops or protein oligomerization. For example, these elements can generate multiple stable states for gene product concentrations, state-dependent responses to stimuli, circadian rhythms and other oscillations, and optimal stimulus frequencies for maximal transcription. In the future, as new experimental techniques increase the ease of characterization of genetic networks, qualitative modeling will need to be supplanted by quantitative models for specific systems.

Classical conditioning of feeding in Aplysia: I. Behavioral analysis.

A training protocol was developed to classically condition feeding behavior in Aplysia californica using tactile stimulation of the lips as the conditional stimulus (CS) and food as the unconditional stimulus (US). Paired training induced a greater increase in the number of bites to the CS than unpaired training or US-only stimulation. Memory for classical conditioning was retained for at least 24 hr. The organization of the reinforcement pathway that supports classical conditioning was analyzed in additional behavioral experiments. No evidence was found for the contribution to appetitive reinforcement of US-mediating pathways originating in the lips of the animals. Bilateral lesions of the anterior branch of the esophageal nerve, which innervates parts of the foregut, however, were found to attenuate classical conditioning. Thus, it appears likely that reinforcement during appetitive classical conditioning of feeding was mediated by afferent pathways that originate in the foregut. The companion paper () describes two neurophysiological correlates of the classical conditioning.

Classical conditioning of feeding in Aplysia: II. Neurophysiological correlates.

Feeding behavior in Aplysia californica can be classically conditioned using tactile stimulation of the lips as conditional stimulus (CS) and food as unconditional stimulus (US) [ (companion paper)]. Conditioning resulted in an increase in the number of CS-evoked bites that persisted for at least 24 hr after training. In this study, neurophysiological correlates of classical conditioning training were identified and characterized in an in vitro preparation of the cerebral and buccal ganglia. Stimulation of a lip nerve (AT(4)), which mediates mechanosensory information, resulted in a greater number of buccal motor patterns (BMPs) in ganglia isolated from animals that had received paired training than in ganglia from control animals. The majority of the evoked BMPs were classified as ingestion-like patterns. Intracellular recordings from pattern-initiating neuron B31/32 revealed that stimulation of AT(4) evoked greater excitatory input in B31/32 in preparations from animals that had received paired training than from control animals. In contrast, excitatory input to buccal neuron B4/5 in response to stimulation of AT(4) was not significantly increased by paired training. Moreover, correlates of classical conditioning were specific to stimulation of AT(4). The number of spontaneously occurring BMPs and the intrinsic properties of two buccal neurons (B4/5 and B31/32) did not differ between groups. These results suggest that appetitive classical conditioning of feeding resulted in the pairing-specific strengthening of the polysynaptic pathway between afferent fibers and pattern-initiating neurons of the buccal central pattern generator.

Modulation of fictive feeding by dopamine and serotonin in aplysia.

The buccal ganglia of Aplysia contain a central pattern generator (CPG) that mediates rhythmic movements of the buccal apparatus during feeding. Activity in this CPG is believed to be regulated, in part, by extrinsic serotonergic inputs and by an intrinsic and extrinsic system of putative dopaminergic cells. The present study investigated the roles of dopamine (DA) and serotonin (5-HT) in regulating feeding movements of the buccal apparatus and properties of the underlying neural circuitry. Perfusing a semi-intact head preparation with DA (50 microM) or the metabolic precursor of catecholamines (L-3-4-dihydroxyphenylalanine, DOPA, 250 microM) induced feeding-like movements of the jaws and radula/odontophore. These DA-induced movements were similar to bites in intact animals. Perfusing with 5-HT (5 microM) also induced feeding-like movements, but the 5-HT-induced movements were similar to swallows. In preparations of isolated buccal ganglia, buccal motor programs (BMPs) that represented at least two different aspects of fictive feeding (i.e., ingestion and rejection) could be recorded. Bath application of DA (50 microM) increased the frequency of BMPs, in part, by increasing the number of ingestion-like BMPs. Bath application of 5-HT (5 microM) did not significantly increase the frequency of BMPs nor did it significantly increase the proportion of ingestion-like BMPs being expressed. Many of the cells and synaptic connections within the CPG appeared to be modulated by DA or 5-HT. For example, bath application of DA decreased the excitability of cells B4/5 and B34, which in turn may have contributed to the DA-induced increase in ingestion-like BMPs. In summary, bite-like movements were induced by DA in the semi-intact preparation, and neural correlates of these DA-induced effects were manifest as an increase in ingestion-like BMPs in the isolated ganglia. Swallow-like movements were induced by 5-HT in the semi-intact preparation. Neural correlates of these 5-HT-induced effects were not evident in isolated buccal ganglia, however.

Computational model of the serotonergic modulation of sensory neurons in Aplysia.

Serotonergic modulation of the sensory neurons that mediate the gill- and tail-withdrawal reflexes of Aplysia is a useful model system for studies of neuronal plasticity that contributes to learning and memory. The effects of serotonin (5-HT) are mediated, in part, via two protein kinases (protein kinase A, PKA, and protein kinase C, PKC), which in turn, modulate at least four membrane currents, including a S ("serotonin-sensitive") K(+) current (I(K, S)), a steeply voltage-dependent K(+) current (I(K-V)), a slow component of the Ca(2+)-activated K(+) current (I(K,Ca-S)), and a L-type Ca(2+) current (I(Ca-L)). The present study investigated how the modulation of these currents altered the spike duration and excitability of sensory neurons and examined the relative contributions of PKA- and PKC-mediated effects to the actions of 5-HT. A Hodgkin-Huxley type model was developed that described the ionic conductances in the somata of sensory neurons. The descriptions of these currents and their modulation were based largely on voltage-clamp data from sensory neurons. Simulations were preformed with the program SNNAP (Simulator for Neural Networks and Action Potentials). The model was sufficient to replicate empirical data that describes the membrane currents, action potential waveform and excitability as well as their modulation by application of 5-HT, increased levels of adenosine cyclic monophosphate or application of active phorbol esters. In the model, modulation of I(K-V) by PKC played a dominate role in 5-HT-induced spike broadening, whereas the concurrent modulation of I(K,S) and I(K,Ca-S) by PKA primarily accounted for 5-HT-induced increases in excitability. Finally, simulations indicated that a PKC-induced increase in excitability resulted from decreases of I(K,S) and I(K,Ca-S), which was likely the indirect result of cross-talk between the PKC and PKA systems. The results provide several predictions that warrant additional experimental investigation and illustrate the importance of considering indirect as well as direct effects of modulatory agents on the modulation of membrane currents.

Effects of macromolecular transport and stochastic fluctuations on dynamics of genetic regulatory systems.

To predict the dynamics of genetic regulation, it may be necessary to consider macromolecular transport and stochastic fluctuations in macromolecule numbers. Transport can be diffusive or active, and in some cases a time delay might suffice to model active transport. We characterize major differences in the dynamics of model genetic systems when diffusive transport of mRNA and protein was compared with transport modeled as a time delay. Delays allow for history-dependent, non-Markovian responses to stimuli (i.e., "molecular memory"). Diffusion suppresses oscillations, whereas delays tend to create oscillations. When simulating essential elements of circadian oscillators, we found the delay between transcription and translation necessary for oscillations. Stochastic fluctuations tend to destabilize and thereby mask steady states with few molecules. This computational approach, combined with experiments, should provide a fruitful conceptual framework for investigating the function and dynamic properties of genetic regulatory systems.

Dopaminergic synapses mediate neuronal changes in an analogue of operant conditioning.

Feeding behavior in Aplysia can be modified by operant conditioning in which contingent reinforcement is conveyed by the esophageal nerve (E n.). A neuronal analogue of this conditioning in the isolated buccal ganglia was developed by using stimulation of E n. as an analogue of contingent reinforcement. Previous studies indicated that E n. may release dopamine. We used a dopamine antagonist (methylergonovine) to investigate whether dopamine mediated the enhancement of motor patterns in the analogue of operant conditioning. Methylergonovine blocked synaptic connections from the reinforcement pathway and the contingent-dependent enhancement of the reinforced pattern. These results suggest that dopamine mediates at least part of the neuronal modifications induced by contingent reinforcement.

In vitro analog of operant conditioning in aplysia. I. Contingent reinforcement modifies the functional dynamics of an identified neuron.

Previously, an analog of operant conditioning in Aplysia was developed using the rhythmic motor activity in the isolated buccal ganglia. This analog expressed a key feature of operant conditioning, namely a selective enhancement in the occurrence of a designated motor pattern by contingent reinforcement. Different motor patterns generated by the buccal central pattern generator were induced by monotonic stimulation of a peripheral nerve (i.e., n.2,3). Phasic stimulation of the esophageal nerve (E n.) was used as an analog of reinforcement. The present study investigated the neuronal mechanisms associated with the genesis of different motor patterns and their modifications by contingent reinforcement. The genesis of different motor patterns was related to changes in the functional states of the pre-motor neuron B51. During rhythmic activity, B51 dynamically switched between inactive and active states. Bursting activity in B51 was associated with, and predicted, characteristic features of a specific motor pattern (i.e., pattern I). Contingent reinforcement of pattern I modified the dynamical properties of B51 by decreasing its resting conductance and threshold for eliciting plateau potentials and thus increased the occurrences of pattern I-related activity in B51. These modifications were not observed in preparations that received either noncontingent reinforcement (i.e., yoke control) or no reinforcement (i.e., control). These results suggest that a contingent reinforcement paradigm can regulate the dynamics of neuronal activity that is centrally programmed by the intrinsic cellular properties of neurons.

In vitro analog of operant conditioning in aplysia. II. Modifications of the functional dynamics of an identified neuron contribute to motor pattern selection.

Previously, an analog of operant conditioning was developed using the buccal ganglia of Aplysia, the probabilistic occurrences of a specific motor pattern (i.e., pattern I), a contingent reinforcement (i.e., stimulation of the esophageal nerve), and monotonic stimulation of a peripheral nerve (i.e., n.2,3). This analog expressed a key feature of operant conditioning (i.e., selective enhancement of the probability of occurrence of a designated motor pattern by contingent reinforcement). In addition, the training induced changes in the dynamical properties of neuron B51, an element of the buccal central pattern generator. To gain insights into the neuronal mechanisms that mediate features of operant conditioning, the present study identified a neuronal element that was critically involved in the selective enhancement of pattern I. We found that bursting activity in cell B51 contributed significantly to the expression of pattern I and that changes in the dynamical properties of this cell were associated with the selective enhancement of pattern I. These changes could be induced by an explicit association of reinforcement with random depolarization of B51. No stimulation of n.2,3 was required. These results indicate that the selection of a designated motor pattern by contingent reinforcement and the underlying neuronal plasticity resulted from the association of reinforcement with a component of central neuronal activity that contributes to a specific motor pattern. The sensory stimulus that allows for occurrences of different motor acts may not be critical for induction of plasticity that mediates the selection of a motor output by contingent reinforcement in operant conditioning.

Control of multistability in ring circuits of oscillators.

The essential dynamics of some biological central pattern generators (CPGs) can be captured by a model consisting of N neurons connected in a ring. These circuits, like many oscillatory nonlinear circuits of sufficient complexity, are capable of multistability, that is, of generating different firing patterns distinguished by the phasic relationships between the firing in each circuit element (neuron). Moreover, a shift in firing pattern can be induced by a transient perturbation. A systematic approach, based on phase-response curve (PRC) theory, was used to determine the optimum timing for perturbations that induce a shift in the firing pattern. The first step was to visualize the solution space of the ring circuit, including the attractive basins for each stable firing pattern; this was possible using the relative phase of N-1 oscillators, with respect to an arbitrarily selected reference oscillator, as coordinate axes. The trajectories in this phase space were determined using an iterative mapping based only on the PRCs of the uncoupled component oscillators; this algorithm was called a circuit emulator. For an accurate mapping of the attractive basin of each pattern exhibited by the ring circuit, the emulator had to take into account the effect of a perturbation or input on the timing of two bursts following the onset of the perturbation, rather than just one. The visualization of the attractive basins for rings of two, three, and four oscillators enabled the accurate prediction of the amounts of phase resetting applied to up to N-1 oscillators within a cycle that would induce a transition from any pattern to any another pattern. Finally, the timing and synaptic characterization of an input called the switch signal was adjusted to produce the desired amount of phase resetting.

Identification and characterization of catecholaminergic neuron B65, which initiates and modifies patterned activity in the buccal ganglia of Aplysia.

Catecholamines are believed to play an important role in regulating the properties and functional organization of the neural circuitry mediating consummatory feeding behaviors in Aplysia. In the present study, we morphologically and electrophysiologically identified a pair of catecholaminergic interneurons, referred to as B65, in the buccal ganglia. Their processes innervate both the ipsi- and contralateral neuropil, and separate branches of B65 appeared to innervate the somata of both ipsi- and contralateral B4/5 neurons. B65 exhibited patterned burst(s) of activity during spontaneous cycles of fictive feeding. Patterned activity in B65 also was elicited by stimulation of the radula nerve, by depolarization of the pattern initiating neurons B31/32 or B63, and by bath application of -3,4-dihydroxyphenylalanine (DOPA). B65 appeared to be a member of the protraction group of neurons. Action potentials in B65 elicited fast one-for-one excitatory postsynaptic potentials (EPSPs) in neurons B4/5, B8A/B, B31/32, B63, and B64. In turn, B31/32 and B63 excited B65 and B64 inhibited B65. Some of the synaptic connections of B65 were plastic. For example, the fast EPSPs elicited in B4/5 and B64 decremented, whereas those in B31/32 andB8A/B facilitated. In addition to fast EPSPs, B65 elicited slow postsynaptic potentials in some of its follower cells. Depolarization of B65 elicited cycles of patterned activity indicative of fictive feeding in buccal neurons, including B65 itself. During series of B65-induced patterns, the properties of the buccal motor programs appeared to change. In particular, the activity of radula closure motor neurons B8A/B, which initially coincided mainly with the protraction phase of a cycle, gradually extended to overlap mostly with the retraction phase. This observation suggests that prolonged activity in B65 may play a role in transitioning from rejection-like to ingestion-like fictive feeding. The phase shift of the activity of B8A/B appears due, at least in part, to a decrease in activity of B4/5, and thus a reduction in inhibition from B4/5 onto B8A/B, during the retraction phase. The functional properties and synaptic connections of B65 suggest that it may play an important role in determining features of patterned neural activity in the buccal ganglia.

Frequency selectivity, multistability, and oscillations emerge from models of genetic regulatory systems.

To examine the capability of genetic regulatory systems for complex dynamic activity, we developed simple kinetic models that incorporate known features of these systems. These include autoregulation and stimulus-dependent phosphorylation of transcription factors (TFs), dimerization of TFs, crosstalk, and feedback. The simplest model manifested multiple stable steady states, and brief perturbations could switch the model between these states. Such transitions might explain, for example, how a brief pulse of hormone or neurotransmitter could elicit a long-lasting cellular response. In slightly more complex models, oscillatory regimes were identified. The addition of competition between activating and repressing TFs provided a plausible explanation for optimal stimulus frequencies that give maximal transcription. Such optimal frequencies are suggested by recent experiments comparing training paradigms for long-term memory formation and examining changes in mRNA levels in repetitively stimulated cultured cells. In general, the computational approach illustrated here, combined with appropriate experiments, provides a conceptual framework for investigating the function of genetic regulatory systems.

Contingent-dependent enhancement of rhythmic motor patterns: an in vitro analog of operant conditioning.

Operant conditioning is characterized by the contingent reinforcement of a designated behavior. Previously, feeding behavior in Aplysia has been demonstrated to be modified by operant conditioning, and a neural pathway (esophageal nerve; E n.) that mediates some aspects of reinforcement has been identified. As a first step toward a cellular analysis of operant conditioning, we developed an in vitro buccal ganglia preparation that expressed the essential features of operant conditioning. Motor patterns that represented at least two different aspects of fictive feeding (i.e., ingestion-like and rejection-like motor patterns) were elicited by tonic stimulation of a peripheral buccal nerve (n.2,3). Three groups of preparations were examined. In a contingent-reinforcement group, stimulation of E n. was contingent on the expression of a specific type of motor pattern (i.e., either ingestion-like or rejection-like). In a yoke-control group, stimulation of E n. was not contingent on any specific pattern. In a control group, E n. was not stimulated. The frequency of the reinforced pattern increased significantly only in the contingent-reinforcement group. No changes were observed in nonreinforced patterns or in the motor patterns of the control and yoke-control groups. Contingent reinforcement of the ingestion-like pattern was associated with an enhancement of activity in motor neuron B8, and this enhancement was specific to the reinforced pattern. These results suggest that the isolated buccal ganglia expressed an essential feature of operant conditioning (i.e., contingent reinforcement modified a designated operant) and that this analog of operant conditioning is accessible to cellular analysis.

Modulation of a cAMP/protein kinase A cascade by protein kinase C in sensory neurons of Aplysia.

The synaptic connections between the sensory neurons of Aplysia and their follower neurons have been used as a model system for examining the cellular mechanisms contributing to neuronal and synaptic plasticity. Recent studies suggest that at least two protein kinases, protein kinase A (PKA) and protein kinase C (PKC), contribute to serotonin (5-HT)-induced short-term facilitation. The interaction between these two kinase cascades has not been examined, however. Using electrophysiological and biochemical approaches, we examined possible interactions between PKA and PKC cascades. The results indicated that prolonged activation of PKC by preincubation with phorbol esters attenuated PKA-mediated actions of 5-HT, including increases in sensory neuron excitability and spike broadening in the presence of tetraethylammonium (TEA) and nifedipine. Although phorbol esters also attenuated increases in excitability by an analog of cAMP and small cardioactive peptide B (SCPB), the degree of attenuation was smaller. In addition, phorbol esters did not attenuate broadening of TEA spikes by the cAMP analog and SCPB. Thus, phorbol esters appeared specifically to attenuate aspects of the 5-HT activation of the cAMP/PKA cascade. Measurements of cAMP levels with radioimmunoassays revealed that phorbol esters did not attenuate 5-HT-induced cAMP synthesis, however. Finally, the results indicated that phorbol esters themselves induced a small but significant increase in excitability as well as an increase in the level of cAMP. Our results suggest that there is crosstalk between the PKC and PKA cascades. The mechanisms by which phorbol esters specifically attenuate 5-HT-induced activation of the cAMP/PKA cascade are not known, however.

Role of calcium-calmodulin-dependent protein kinase II in modulation of sensorimotor synapses in Aplysia.

The Ca2+-calmodulin-dependent protein kinase II (CaMKII) inhibitor, [1-[N,O-bis(5-isoquinolinesulfonyl)-N-methyl-L-tyrosyl]-4-phenylpiperazi ne) (KN-62), was used to investigate the role of CaMKII in synaptic transmission and serotonin (5-HT)-induced facilitation in Aplysia. Application of KN-62 (10 microM) by itself increased the amplitude of excitatory postsynaptic potentials (EPSPs) at sensorimotor synapses in pleural-pedal ganglia. Moreover, in the presence of KN-62, 5-HT-induced short-term facilitation was attenuated. Application of KN-62 by itself slightly increased the duration of action potentials in isolated sensory neuron somata but did not block spike broadening produced by 5-HT. KN-62 had no effect on excitability of isolated sensory neuron somata nor did it block 5-HT-induced enhancement of excitability. These results indicate that the attenuation of short-term facilitation by KN-62 is not due to modulation of the membrane currents contributing to 5-HT-induced spike broadening or enhancement of excitability. Rather, these data are consistent with the hypothesis that CaMKII contributes to the regulation of sensorimotor connections and that it has a role in spike-duration-independent processes contributing to short-term facilitation.

Differential effects of 4-aminopyridine, serotonin, and phorbol esters on facilitation of sensorimotor connections in Aplysia.

Serotonergic modulation of sensory neurons in Aplysia and their synaptic connections with follower cells has been used extensively as a model system with which to study mechanisms underlying neuronal plasticity. Serotonin (5-HT)-induced facilitation of sensorimotor connections is due to at least two processes: a process related to the broadening of presynaptic action potentials and a spike-duration-independent (SDI) process that may involve mobilization of transmitter. We have examined the relationship between spike broadening and synaptic facilitation of relatively nondepressed sensorimotor connections in the intact pleural-pedal ganglia. Previously, 5-HT-induced spike broadening in the sensory neuron was shown to be primarily due to the modulation of a voltage-dependent K+ current (Ik.v). Low concentrations (20-30 microM) of 4-aminopyridine (4-AP) were used to rather selectively block Ik.v. 4-AP increased spike duration in the sensory neuron and the excitatory postsynaptic potential (EPSP) in the motor neuron. The temporal development of 4-AP-induced spike broadening closely parallel that of synaptic facilitation. Thus spike broadening via the reduction of Ik.v can directly contribute to synaptic facilitation. The relationship between spike broadening induced by 5-HT (10 microM) and enhancement of the EPSP was also analyzed. We found that components of 5-HT-induced synaptic facilitation preceded the development of 5-HT-induced spike broadening. The comparison between the results of 4-AP and 5-HT revealed that the SDI processes made an important contribution to the rapid development of 5-HT-induced synaptic facilitation and that spike broadening made an important contribution to its maintenance. The SDI process and a slowly developing component of 5-HT-induced spike broadening are mediated, at least in part, by the activation of protein kinase C (PKC). Application of phorbol 12,13-diacetate (PDAc), an activator of PKC, partially mimicked the effects of 5-HT on spike duration and the EPSP. PDAc-induced enhancement of the EPSP preceded the slower development of PDAc-induced spike broadening. Like 5-HT, PDAc enhanced the EPSP via both spike broadening and the SDI processes. In addition, a 15-min exposure to PDAc occluded 5-HT-induced enhancement of the EPSP, suggesting that PKC and 5-HT engage similar or overlapping mechanisms. On the basis of these results and others, we propose a time-dependent hypothesis for the 5-HT-induced synaptic facilitation of nondepressed synapses, in which multiple second-messenger/protein kinase systems mediate the actions of 5-HT via both spike-duration-dependent and SDI processes.

Phase response characteristics of model neurons determine which patterns are expressed in a ring circuit model of gait generation.

In order to assess the relative contributions to pattern-generation of the intrinsic properties of individual neurons and of their connectivity, we examined a ring circuit composed of four complex physiologically based oscillators. This circuit produced patterns that correspond to several quadrupedal gaits, including the walk, the bound, and the gallop. An analysis using the phase response curve (PRC) of an uncoupled oscillator accurately predicted all modes exhibited by this circuit and their phasic relationships--with the caveat that in certain parameter ranges, bistability in the individual oscillators added nongait patterns that were not amenable to PRC analysis, but further enriched the pattern-generating repertoire of the circuit. The key insights in the PRC analysis were that in a gait pattern, since all oscillators are entrained at the same frequency, the phase advance or delay caused by the action of each oscillator on its postsynaptic oscillator must be the same, and the sum of the normalized phase differences around the ring must equal to an integer. As suggested by several previous studies, our analysis showed that the capacity to exhibit a large number of patterns is inherent in the ring circuit configuration. In addition, our analysis revealed that the shape of the PRC for the individual oscillators determines which of the theoretically possible modes can be generated using these oscillators as circuit elements. PRCs that have a complex shape enable a circuit to produce a wider variety of patterns, and since complex neurons tend to have complex PRCs, enriching the repertoire of patterns exhibited by a circuit may be the function of some intrinsic neuronal complexity. Our analysis showed that gait transitions, or more generally, pattern transitions, in a ring circuit do not require rewiring the circuit or any changes in the strength of the connections. Instead, transitions can be achieved by using a control parameter, such as stimulus intensity, to sculpt the PRC so that it has the appropriate shape for the desired pattern(s). A transition can then be achieved simply by changing the value of the control parameter so that the first pattern either ceases to exist or loses stability, while a second pattern either comes into existence or gains stability. Our analysis illustrates the predictive value of PRCs in circuit analysis and can be extended to provide a design method for pattern-generating circuits.