Asymmetric and transient properties of reciprocal activity of antagonists during the paw-shake response in the cat

Mutually inhibitory populations of neurons, half-center oscillators (HCOs), are commonly involved in the dynamics of the central pattern generators (CPGs) driving various rhythmic movements. Previously, we developed a multifunctional, multistable symmetric HCO model which produced slow locomotor-like and fast paw-shake-like activity patterns. Here, we describe asymmetric features of paw-shake responses in a symmetric HCO model and test these predictions experimentally. We considered bursting properties of the two model half-centers during transient paw-shake-like responses to short perturbations during locomotor-like activity. We found that when a current pulse was applied during the spiking phase of one half-center, let’s call it #1, the consecutive burst durations (BDs) of that half-center increased throughout the paw-shake response, while BDs of the other half-center, let’s call it #2, only changed slightly. In contrast, the consecutive interburst intervals (IBIs) of half-center #1 changed little, while IBIs of half-center #2 increased. We demonstrated that this asymmetry between the half-centers depends on the phase of the locomotor-like rhythm at which the perturbation was applied. We suggest that the fast transient response reflects functional asymmetries of slow processes that underly the locomotor-like pattern; e.g., asymmetric levels of inactivation across the two half-centers for a slowly inactivating inward current. We compared model results with those of in-vivo paw-shake responses evoked in locomoting cats and found similar asymmetries. Electromyographic (EMG) BDs of anterior hindlimb muscles with flexor-related activity increased in consecutive paw-shake cycles, while BD of posterior muscles with extensor-related activity did not change, and vice versa for IBIs of anterior flexors and posterior extensors. We conclude that EMG activity patterns during paw-shaking are consistent with the proposed mechanism producing transient paw-shake-like bursting patterns found in our multistable HCO model. We suggest that the described asymmetry of paw-shaking responses could implicate a multifunctional CPG controlling both locomotion and paw-shaking.


Introduction
We engage in many types of rhythmic motor behaviors in our everyday lives, such as walking, breathing, chewing, etc. Rhythmic behaviors like these are generally controlled by interneuronal networks called central pattern generators (CPGs) [1][2][3][4]. Many of these behaviors are steady-state and long-lasting, such as walking and breathing. Other rhythmic behaviors tend to last for only a short interval of time and may have variable characteristics, such as scratching, struggling, gasping, etc. Although a wide variety of motor rhythms have been described in the literature, the mechanisms underlying these rhythms are still debated. For example, the relatively slow locomotor rhythm and the much faster paw-shake response in the cat could be controlled by two separate independent CPGs, separate CPGs with shared interneuronal circuits, or a single multifunctional CPG. There is experimental evidence from a variety of species supporting the possibility of multifunctional CPGs controlling a variety of different motor behaviors [5][6][7][8][9][10][11][12][13].
In principle, multifunctional CPGs could produce multiple rhythms of different frequencies. Relatively slow long-lasting rhythmic behaviors could be controlled by a stable, steadystate rhythm of a multifunctional CPG until an input signal changes or halts its rhythmic activity. A short-duration, non-steady-state rhythmic behavior, on the other hand, could result from transient neural dynamics of the same CPG triggered by a perturbation, e.g., a sensory input signal. These transient neural dynamics could generate a short-duration rhythmic behavior as a reliable response to certain environmental stimuli, such as transient responses seen in insect olfaction processing [14][15][16].
We have recently developed a multifunctional CPG model that generates reciprocal rhythmic activity between cat hindlimb antagonists and exhibits multistability of slow, locomotorlike and fast, paw-shake-like steady-state rhythms [17,18]. The model is based on a novel mechanism that produces multistability of two rhythms that differ in frequency by an order of magnitude, e.g., 1 Hz and 10 Hz. This mechanism involves two slow, intrinsic currents in each half-center neuron with significantly different kinetics of inactivation. These two currents are slowly inactivating, low-voltage-activated calcium current (I CaS ) and slowly inactivating sodium current (I NaS ). The locomotor-like rhythm in our model is largely generated and controlled by I CaS , while the paw-shake-like rhythm is largely generated and controlled by I NaS .
Although I CaS and I NaS both inactivate relatively slowly, the time constant of inactivation of I CaS is much larger than the time constant of inactivation of I NaS . The slow calcium current in our model deinactivates at voltages that are more hyperpolarized than the minimum voltage reached during the stable paw-shake-like rhythm. Therefore, I CaS is completely inactivated during the paw-shake-like rhythm, allowing I NaS to control the paw-shake-like rhythm and maintain a much higher frequency than the locomotor-like rhythm. During the locomotorlike rhythm, however, I CaS and I NaS are both active, but the kinetics of I CaS bring the rhythm to a much slower frequency. With a short perturbation (a pulse of conductance) to one or both half-center neurons, the model can be switched from the locomotor-like rhythm to the pawshake-like rhythm or vice versa. We have shown that a neuromechanical model of cat hindlimbs controlled by this multifunctional CPG and motion-dependent somatosensory feedback could reproduce not only distinct rhythms of cat locomotion and paw-shaking but also the typical reciprocal activation of flexors and extensors during locomotion and the atypical reciprocal activation of anterior and posterior muscles during paw-shaking [17,19]. In a previous study, we modeled steady-state paw-shake-like activity patterns [18], whereas recorded paw-shake cycle periods are not steady but become longer with each subsequent paw-shake cycle [19]. This suggests that paw-shake responses could be generated by transient dynamics elicited by some sensory stimuli. Here, we investigated potential mechanisms that can produce transient paw-shake-like responses to short pulses of current applied at various phases of locomotor-like activity of our previously developed multistable, half-center oscillator CPG model [17,18]. We also compared characteristics of the model's transient dynamics with the characteristics of EMG activity recorded during paw-shake responses in unrestrained cats.
Thus, the first goal of this study was to investigate the mechanism by which neural transient dynamics could be produced in the multistable, half-center oscillator CPG model. The second goal was to investigate temporal characteristics of EMG activity of cat hindlimb antagonistic muscles (durations of activity cycles, bursts and interburst intervals) in consecutive paw-shake cycles evoked during locomotion and compare them with those of the model. We tested the hypothesis that the temporal characteristics of recorded EMG activities during cat paw-shake responses would be consistent with the model transient dynamics.

Ethics statement
All surgical and experimental procedures of this study were in compliance with the "Guide for the Care and Use of Laboratory Animals. Eighth Edition" (National Research Council, 2011) and were approved by the Institutional Animal Care and Use Committee of the Georgia Institute of Technology (protocol number A13063).

Modeling
We used our previously published Hodgkin-Huxley style model [17,18] to investigate fast paw-shake-like activity as a transient response to a perturbation of a steady-state locomotorlike rhythm. Our model consists of two identical, mutually inhibiting neurons with the same intrinsic currents and the same synaptic currents. There are five intrinsic currents in each neuron: a fast sodium current (I NaF ), a slowly inactivating sodium current (I NaS ), a potassium current (I K ), a low-voltage-deinactivated, slowly inactivating calcium current (I CaS ), and a leak current (I leak ). Currents I NaF and I K were modified from another mammalian locomotion CPG model [20,21]. The following equations use the units of second (s), milliVolt (mV), nanoSiemens (nS), and picoAmpere (pA): NaF;1 h NaF ½V À E Na � and inactivation variables, respectively, of some current I x . The activation and inactivation variables range from 0 to 1, where the current is fully deactivated when m x = 0 and fully activated when m x = 1; the current is fully inactivated when h x = 0 and is fully deinactivated when h x = 1 [22]. The variable m SynPre is the synaptic activation variable of the presynaptic neuron which is governed by the corresponding presynaptic membrane potential V Pre . The capacitance C in this model is 0.001 nF.
We integrated these equations using the 8th order Runge-Kutta integration method and GSL ODE solver software, with a maximum step size of 10 −5 seconds, an absolute tolerance of 10 −8 and relative tolerance of 10 −9 . Values of the model parameters are provided in Tables 1  and 2. The transient fast rhythmic activity was elicited in our model by applying a pulse of an additional excitatory modulatory current, I E with an excitatory reversal potential E E = 0 mV, as a square pulse of conductance, g E . This pulse of modulatory current was applied to both neurons in the model when the model was engaged in a slow locomotor-like rhythm. It was applied at a specified time, t pul , corresponding to a specific phase, p pul , and it continued for a specified duration, d pul . The fast rhythmic activity that results from this pulse may be either transient or stable. Many cases of transient paw-shake-like activity were collected by varying the time of pulse onset, t pul , and the pulse duration, d pul . The phase p pul = 0% was defined as the instant of the first spike peak in each burst of neuron 2. We varied p pul from 0% to 100% with a step of 0.25% of the cycle, and for each of these phases, we varied pulse duration d pul from 200 ms to 1200 ms with a step of 5 ms.
For each episode of transient paw-shake-like activity in the model, we measured the temporal characteristics of bursting activities of the two neurons comprising the half-center oscillator CPG for each subsequent activity cycle. These characteristics included bursting cycle period (CP), burst duration (BD), and interburst interval duration (IBI). CP of each neuron was calculated as the time between the peak of the first spike of a burst and the peak of the first spike of the next burst. BD was calculated as the time between the peak of the first spike of a burst and the peak of the last spike of the same burst. IBI was defined as (CP-BD). Duty cycle (DC) was defined as 100 � BD/CP. Paw-shake-like rhythms were defined as rhythms with CPs < 210 ms. The progressions of BD and IBI throughout each episode of transient activity were measured for each neuron and each phase of pulse onset. Linear regression analysis was used to perform a linear curve fit in order to evaluate the rate of change of BD and IBI over time during transient activity. We excluded the very first burst after the pulse from the linear curve fit, because there were some simulations where the first burst was very short, sometimes consisting of only 2 or 3 short spikes. For each 0.25%-phase increment of pulse onset, we obtained a simulation of transient paw-shake-like activity. Among all collected responses, we selected and used for further analyses those which contained between 5 and 9 paw-shake-like burst cycles, and disregarded simulations with fewer or greater. We applied linear curve fits to BD and IBI versus time for each collected simulation, determined the slopes of these fits for each simulation, and used a sliding averaging of the slopes with a window width of 1.25% of the cycle duration. That gave us the mean slope values of BD and IBI for all simulations corresponding to each perturbation phase. We also investigated more closely the mean slopes of BD and IBI in the phase range of 20% -40% which approximately corresponds to the locomotor cycle phases in which paw-shaking is actually initiated in cats. We hypothesized that the asymmetry of the linear trends of BD and IBI throughout time in our model could be explained by the functional antiphase relationships between the slowest state variables of the two half-centers. To test this hypothesis, we built a reduced version of the model, which we called the "constant h CaS model". In each neuron of the reduced model, the h CaS state variable was held constant and its corresponding equations were removed from the CPG model. We separately varied h CaS1 and h CaS2 as two parameters, producing a simulation of the model for each combination of values in a certain domain. Each simulation was integrated for 1000 s to ensure that the activity had reached a steady state. We obtained the mean values of BD and IBI for neurons #1 and #2 separately by averaging over the last 20 bursts for each simulation case. We measured the difference of DC between the two model neurons and plotted this difference versus the difference in h CaS of the two neurons, and we performed a linear curve fit on these data.

Animal experiments
We analyzed EMG recordings from 12 adult female cats (mass 3.56±0.73 kg) that were part of our previous studies [23][24][25][26]. Details of surgical and experimental procedures were described previously [23,24]. Briefly, cats were trained with food rewards to walk on a plexiglass enclosed walkway. After training, the major ankle, knee and hip muscles of one hindlimb were implanted with bipolar wire electrodes (CW5402; Cooner Wire, Chatsworth, CA, USA) under aseptic conditions and general isoflurane anesthesia (1-3%). The animals recovered after surgery for 2 weeks. Pain medication (fentanyl transdermal patch, 12-25 μg/h and/or buprenorphine, s.c., 0.01 mg/kg, or ketoprofen, 2 mg/kg, s.c.) and antibiotics (cefovecin, 8 mg/kg, s.c., or ceftiofur, 4 mg/kg, s.c.) were administered for 3 and 10 days, respectively. To evoke pawshake responses during walking, we attached a small piece of adhesive tape (~2 cm x 3 cm) to paw pads of the implanted hindlimb and placed the cat on the walkway. The animal walked across the walkway and performed paw-shaking during the swing phase of the hindlimb keeping the other limbs on the ground until termination of paw-shaking and resuming locomotion [23,27]. EMG activity was recorded at a sampling rate of 3000 Hz. Recorded EMG signals were band pass filtered (30-1000 Hz) and amplified. Since our CPG model describes the operation of a HCO that acts as a rhythm generator and generates excitatory inputs to the neural network that activates flexor and extensor motoneurons, we analyzed paw-shake EMG recordings from hindlimb muscles with flexor and extensor locomotor activity (25). These muscles included flexors tibialis anterior (TA, ankle flexor) and iliopsoas (IP, hip flexor) and extensors soleus (SO, ankle extensor), medial gastrocnemius (MG, ankle extensor and knee flexor), vastus lateralis and vastus medialis (VA, knee extensor), and anterior biceps femoris (BFA, hip extensor).
For each of these muscles, we measured the progression of EMG burst characteristics of paw-shake cycles throughout each paw-shake episode. In order to measure these burst characteristics, we first full-wave rectified the EMG signal and slightly smoothed it using a zero-lag, moving average with an 8-ms window. For each channel, we then took a random time interval with no muscle activity that was at least 200 ms wide and determined the mean and standard deviation of the raw EMG signal within this selection. The standard deviation was used as a measure of the EMG noise level in each trial. We set the activity burst threshold to be 3 times this standard deviation. We defined the EMG paw-shake cycle period (CP) for each muscle as the time between the EMG burst onset in a given cycle and the burst onset in the next cycle. The EMG burst duration BD was defined as the time between the burst onset and offset, and EMG interburst interval IBI was defined as the time between the burst offset and next burst onset. The duty cycle DC was calculated as 100 � BD/CP. We analyzed changes in BD and IBI in all consecutive cycles of individual paw-shake episodes for each muscle: 28 episodes for MG, 28 for SO, 12 for VA, 12 for BFA, 21 for IP, and 12 for TA (Table 3). Each paw-shake episode had between 5 and 11 cycles. We calculated Pearson's correlation coefficient for BD and IBI versus time for each paw-shake response within each muscle group. The median Pearson's correlation coefficient (R) of all trials within each muscle group was greater than 0.7 for IBI of MG, SO, VA, and BFA; and for BD of IP, TA, and VA. For the IBI of IP and TA and for BD of MG, SO and BFA, BD/IBI appeared to remain constant over time, and therefore, R varied widely from trial to trial with a median R anywhere  from -0.3 to 0.3. We used linear regression analysis of BD and IBI versus time for each pawshake response to see if the median curve fit slopes were statistically different from zero in the case of variables that appeared to remain constant and to quantify the mean and median rates of change of these variables over time in the case of groups with high R values. In order to be consistent with the model analysis, we excluded the first burst from the regression analysis. We then calculated the slopes of these regression lines for each muscle and paw-shake episode. We determined that we could not assume the regression slopes were normally distributed based on histograms of the slope data and eyeball metrics. We used the one-sample nonparametric Wilcoxon rank test to determine whether the median linear regression slopes were significantly different from zero.

Transient dynamics of half-center oscillator model
In our model of a half-center oscillator, two steady-state oscillatory regimes-slow locomotorlike and fast paw-shake-like, coexist ( Fig 1A and 1B), as previously described in detail in [17,18]. In previous studies, we simulated a paw-shake response during locomotion by applying a pulse of current to switch from stable locomotor-like activity to stable paw-shake-like activity and then applying another pulse of current to switch back to locomotor-like activity.
Here, we reduced the duration of the first pulse in order to produce a transient paw-shake-like response, where a second pulse is not needed to return to locomotor-like activity ( Fig 1C). In this case, the model produced a few paw-shake-like bursts and then slowly (within several seconds) returned to a slow steady-state locomotor-like activity. During this transient activity, BD and IBI of both neurons changed as a sigmoid function of time ( Fig 1C). The duration of the transient paw-shake-like activity (i.e., the number of transient bursts) depended on the phase of pulse onset and the pulse duration (Fig 2). The pulse duration needed to produce this transient activity also changes depending on the phase of pulse onset (Fig 2). In large regions of phase space, which will be explored further in the following, we found asymmetric trends across the two half-centers within transient paw-shake-like activity, despite that our HCO CPG model had two completely symmetric half-centers (two neurons with the same intrinsic and synaptic properties) with identical burst characteristics for the steady-state slow regime and the fast regimes and that both neurons received the same perturbations at the same time. Looking back at the example simulation in Fig 1, when we zoomed in on transient paw-shake-like activity, we found that the burst characteristics of the two neurons evolved asymmetrically throughout the transient paw-shake-like activity (Fig 3). If the pulse was applied during the spiking phase of the locomotor-like burst of neuron #2, then its BD (BD 2 ) grew faster than the BD of neuron #1 (BD 1 ), and the IBI of neuron #1 (IBI 1 ) grew faster than the IBI of neuron #2 (IBI 2 ) ( Fig 3B). For each simulation of transient paw-shake-like activity, we computed linear regressions between BD and the time of burst occurrence within the pawshake-like response, i.e. the burst onset time, between IBI and the burst onset time, and between DC and burst onset time.
We then collected all simulations from the analysis in Fig 3 that had between 5 and 9 pawshake cycles, which amounted to 1328 model simulations relatively evenly distributed across phase of pulse onset ( Fig 4A). In this data set, we found that the regression slopes (the mean rate of change) of BD and IBI for neurons #1 and #2 depended on the phase of pulse onset and pulse duration (Fig 4A). There was substantial variability in the slopes for BD and IBI as evident from Fig 4B, so we used a sliding window method to produce a relatively smoothed, averaged representation of the slopes of BD and IBI versus phase of pulse onset disregarding pulse duration (Fig 4B). At around phase = 10% (near the burst onset of neuron #2), BD 2 slope became greater than BD 1 slope, and at around phase = 55% (near the burst onset of neuron #1), BD 1 slope became greater than BD 2 slope. On the other hand, at around phase = 15%, IBI 1 slope became greater than IBI 2 slope, and at around phase = 60%, IBI 2 slope became greater than IBI 1 slope.

Asymmetric responses of the symmetric HCO model
We then looked more closely at a range of pulse application phases from 20% to 40% where behaviors of BD and IBI in the model and cat experiments were similar (see next section) and where actual paw-shake responses during cat locomotion begin, i.e. in the flexor (swing) phase of a locomotion cycle assuming neurons #1 and #2 represent the extensor and flexor half-centers, respectively. Within this range, the pulse was applied around the middle of the spiking activity of neuron #2, which would represent the swing phase of a cat's locomotion cycle. We produced 286 simulations in this phase range by modifying the phase of pulse onset and the pulse duration, so that the pulse duration ranged from 700 ms to 1000 ms. Slightly varying these pulse parameters produced highly variable effects on the resulting transient activity of  neurons #1 and #2 (Figs 4 and 5). We calculated the mean and median slopes of the regression lines for all of the simulations in this range and found that the median slopes of the linear regression lines for BD, IBI and DC as functions of the burst onset time during paw-shake-like activity were significantly different from zero for neuron #1 and neuron #2. However, the median BD slope of neuron #1 (0.0043) was significantly smaller than the median BD slope of neuron #2 (0.013). The median IBI slope of neuron #2 (0.0037) was significantly smaller than the median IBI slope of neuron #1 (0.011). Moreover, the DC slopes for neurons #1 (-0.0015) was negative, while the DC slope for neuron #2 was positive (0.003) (Fig 5). Thus, despite the identical properties of the two half-centers of the HCO model, they produced asymmetric transients in response to identical perturbation pulses.
We then investigated possible causes of the asymmetric changes of BD and IBI in our symmetric half-center oscillator CPG model. We looked at the differences in inactivation of the two slow currents, h NaS (time constant of 100 ms) and h CaS (time constant of 485 ms) between the two neurons throughout the transient paw-shake-like response (Fig 6). Although values of h NaS at the onset of each burst were asymmetric for the first couple bursts, they did not exhibit substantial asymmetry throughout the entire duration of transient activity and thus could not explain the differences in temporal evolution of BD and IBI between the two neurons ( Fig 6B  and 6D). On the other hand, the values of h CaS of neurons #1 and #2 progressed differently throughout the entire duration of transience (Fig 6C). Although the difference in the h CaS  values at burst onset between the two neurons was small (Fig 6D), we hypothesized that it could explain the asymmetric evolution of BDs and IBIs throughout the transient activity.
While h NaS may play some role in the initial asymmetry in BD and IBI across the two neurons, mainly at the beginning of transient activity, we focused on the role of h CaS due to the clear asymmetry in the progression of h CaS throughout the entire duration of transient-paw-shakelike activity (Fig 6C and 6D). was treated as a parameter, and the corresponding variable and its differential equation was removed from the neuron model. These new parameters, h CaS1 and h CaS2 , corresponding to the inactivation of slow calcium current in neurons #1 and #2, respectively, were kept constant with respect to time. We varied h CaS1 and h CaS2 separately, to produce a simulation of the model for each combination of values. We first kept h CaS2 constant at 0.01, and varied h CaS1 from 0.0075 to 0.02 by steps of 0.0005. We found that BD of neuron #1 increased with the parameter value of h CaS1 , while IBI of neuron #1 increased only slightly as h CaS1 increased ( Fig  7A and 7B). On the other hand, BD of neuron #2 only slightly increased as h CaS1 increased, while IBI of neuron #2 increased significantly as h CaS1 increased (Fig 7A and 7B). We   (Fig 7C). Then for each value of h CaS1 , we varied h CaS2 between 0.0075 and 0.02 with a step size of 0.0005, and we used the same range of values and step size to vary h CaS1 . We found that within this data set of 676 simulations the difference in duty cycle between the two neurons depended linearly on the difference in h CaS between the two neurons in the reduced model (Fig 7D). This means that the difference in BD between the two neurons normalized by their shared cycle period depends linearly on the difference in h CaS of the two neurons. Therefore, the duty cycles are directly and predictably controlled by the difference in h CaS (h CaS2 -h CaS1 ) between the two neurons. We concluded that the asymmetry in transient responses in the full model occurred due to the difference in how much the slow calcium currents in both neurons inactivate while they are depolarized during the pulse.

Fig 7. Dependence of burst duration (BD) and interburst interval (IBI) in model HCO neurons #1 and #2 on h CaS1 . (A)
In order to determine whether h NaS plays any notable role in generating the asymmetric trends throughout paw-shake transience, we set h CaS constant in both neurons at the moment the pulse ended using the same example of transient activity that was used in Figs 1, 3, and 6 ( Fig 8A). At the moment the pulse ended, h CaS1 and h CaS2 were held constant at the values they had reached, and thereby, at different asymmetric values (h CaS1 = 0.0311, h CaS2 = 0.0242) (Fig  8). We found that while the baseline values for BD and IBI were asymmetric across the two model neurons, they remained roughly constant at those values indefinitely (Fig 8B). We applied this same procedure to all of the cases of transience previously analyzed in 20% -40% phase range and found that this result was consistent (Fig 8E and 8G). To quantify this result, we applied linear regression analysis to the first few bursts after the pulse, where the number of bursts included was constrained to the number of bursts in the original transient paw-shakelike activity of the corresponding simulation. Since the asymmetric trends in BD and IBI disappear when h CaS is held constant, the kinetics of the inactivation of slow calcium current is the sole driver of these asymmetric trends in our model.
We then performed the same procedure except that we held h CaS constant at the same value for both neurons, namely at the average of the two values reached at the moment of pulse offset (Fig 8C, 8D, 8F, and 8H). We found no asymmetric trends of BD and IBI, and we found that the asymmetry in the baseline values for BD and IBI disappeared; BD and IBI both remained roughly constant at the same value for both neurons (Fig 8D, 8F, and 8H). Therefore, the kinetics of the inactivation of slow calcium current is the sole driver of asymmetry in the baseline values of BD and IBI during transient paw-shake-like activity.

Temporal dynamics of EMG activity bursts during experimental paw-shake response
During actual paw-shaking in cats, the recorded muscles demonstrated reciprocal extensor and flexor EMG activity (Fig 9A and 9B). The paw-shake CP progressively increased at each consecutive cycle within an episode (Figs 9C and 10), consistent with a previous study [19]. Since CP grows throughout a paw-shake response, one might expect that BD and IBI would both grow throughout a paw-shake response also. That was not the case for most of the muscles we recorded. BDs of flexors (TA and IP) tended to grow throughout a response, while flexor IBIs did not; on the other hand, IBIs of extensors (SO, MG, VA, and BFA) tended to grow throughout a response, while extensor BDs did not, except for VA (Figs 9C and 10). This unexpected asymmetry may offer some insight into the neural dynamics that could produce this rhythmic response. Interestingly, we also found that flexor EMG activity had a greater DC  (Figs 9A and 9B and 10).
After using the same Wilcoxon signed rank test on slopes of EMG bursts of flexor muscles IP and TA (Figs 9 and 10), we obtained mostly opposite results. Specifically, the median BD slopes were significantly positive for these muscles (Figs 10 and 11A; for IP, p = 2.1x10 -4 , n = 21 and for TA, p = 4.9x10 -4 , n = 12), while the IBI slopes were not significantly different from zero for IP and TA (Figs 10 and 11B; for IP, p = 0.357, n = 21; and for TA, p = 0.151, n = 12).
The DC slope for IP was significantly greater than zero (p = 0.0496, n = 21). The median DC slope for TA was positive but not significantly different from zero (p = 0.677, n = 12).

Discussion
In this study, we demonstrated that the model of a multistable half-center oscillator CPG, capable of producing steady-state slow locomotor-like and fast paw-shake-like rhythms [17,18], could also produce a fast paw-shake-like rhythm as a transient response to a perturbation of the slow steady-state locomotor-like rhythm (Fig 1). The transient activity was asymmetric with different evolution rates for BD, IBI, and DC of the two half-centers throughout the transient response, despite that the two model half-centers are entirely identical (Figs 3-5). In the model, we determined that this asymmetry resulted from the anti-phasic activity occurring at the time identical current pulses were applied to both neurons. More specifically, the asymmetry resulted from the different levels of inactivation of the slow calcium currents between the two neurons at the time of pulse offset and the different progressions of these levels throughout the transient activity (Figs 6 and 7).
We also recorded EMG activity of selected hindlimb muscles while cats performed pawshaking during walking. Measured paw-shake cycle periods increased in consecutive pawshake cycles (Figs 9 and 10) in accordance with previous studies [19]. We demonstrated for the first time that EMG burst duration and interburst interval evolve asymmetrically in consecutive paw-shake cycles of hindlimb muscles with flexor-and extensor-related activity (Figs 9-11). This asymmetry was similar to the transient dynamics of the model CPG half-centers (Figs 3 and 5). In the next sections, we discuss these findings in the light of the potential role of multistable CPGs and their possible operation during locomotion and paw-shaking in the cat.

Functional transient activity in biological systems
Central pattern generators are often modeled to produce stable steady-state rhythmic patterns, and each such pattern is an attractor state of the system. When a system exhibiting a steadystate pattern, e.g. a locomotor pattern, is perturbed, some transient activity may occur before the system returns to that attractor state and resumes its steady-state rhythmic pattern [14,15,28]. This transient activity is often not considered in modeling studies, although it may represent functional activity of the network with properties necessary for producing an appropriate response to a stimulus [14,15,[28][29][30].
Studies have shown in various neural circuits of different organisms that transient activity can have functional purposes sometimes more so than steady state activity. For example, olfactory responses are encoded by transient dynamics in many investigated species [14,15,31,32]. In locusts, when an odor is presented in a short puff of air, transient oscillations of membrane potential occur in specific neurons of the antennal lobe depending on the odor presented [31]. Each neuron responds specifically to a small set of odors, and the transient pattern of the evoked oscillation is specific to the odor [31]. If the stimulus is present for more than 2-3 seconds, then the oscillations reach a stable attractor state, and odor specificity decreases during this stable activity [14,31]. The above example demonstrates that the properties of the transient responses to short stimuli may be more important for functionality than the steady-state pattern in locust olfaction. Transient dynamics are known to produce reliable responses to specific stimuli in other systems as well, such as mammalian learning and memory processes and cortical visual processing [15,16,33]. We propose that a mechanism similar to the one we have modeled here could apply to other species where a brief behavior is required to respond to some environmental stimulus. For instance, the struggling behavior in Xenopus tadpoles and larval zebrafish [6,11,12,34] and the scratching behavior in turtles [5,6,35,36]. Struggling and scratching are transient, reflexive behaviors that could be generated by a transient response of the swimming CPG in either organism.

Potential transient dynamics of locomotor CPG controlling the paw-shakelike pattern
Given that the recorded paw-shake cycle periods change in each successive cycle (Figs 9-11 and [19]), we propose that cat paw-shaking may result from a locomotor CPG's transient activity. Paw-shaking can be elicited in deafferented spinalized cats [37] and fictive paw-shake-like responses can be produced in spinal-transected curarized cat preparations [38], suggesting that the paw-shaking rhythm is generated by a spinal CPG. Although motion-dependent somatosensory feedback affects the centrally generated paw-shake activity patterns, these effects appear to be limited to two muscle groups responsible for the atypical muscle synergy during paw-shaking-the vasti (VA) and tibialis anterior (TA) muscles [37].
In the current study, we investigated the question of whether paw-shake responses can be produced as a transient response of the multifunctional locomotor CPG. Our HCO CPG model can exhibit transient paw-shake-like activity in two different types of parameter regimes, (1) when the model exhibits coexistence of paw-shake-like activity and locomotorlike activity, and (2) when the model exhibits only locomotor-like activity but is set nearby the stable paw-shake-like regime in parameter space. In (2), having CPG parameters set in the vicinity of the range supporting the fast paw-shake-like regime as an attractor relates to the concept of a conditional burster: a neuron producing endogenous bursting activity under certain physiological conditions, for example under certain neuromodulatory tone. If both slow and fast regimes are attractors as in case (1) then transient paw-shake-like activity could be induced due to a slow passage near basin of attraction of the fast regime as a result of perturbation which shifted trajectory of HCO activity close to the border between basins of attraction of the two regimes but did not make it cross the border. Our model uses a relatively novel mechanism for multistability of bursting regimes with largely distinct periods that involves the different kinetics of inactivation of two intrinsic slow currents [17]. These two slow currents were built with largely distinct time constants of inactivation and different voltages of halfinactivation in order to produce two rhythmic bursting regimes with cycle periods that differ by an order of magnitude, the locomotor regime (1 Hz) and the paw-shake-like regime (10 Hz). The model was built so that the kinetics of inactivation of the slow Ca ++ (I CaS ) and Na + (I NaS ) currents would drive the kinetics of inactivation of the slow and fast regime, respectively. When transient activity is generated in our model, slow Na + current is still the main driving current of the transient fast activity, but slow Ca ++ current controls the way burst duration and interburst interval change over time and the way the transient activity terminates and returns to the slow rhythm. The transient rhythmic bursting is reliably controlled by the slowest variable of the system, h CaS , which drives the slow bursting rhythm, locomotor-like bursting.
While it has been shown in other CPG models that asymmetric external input can create asymmetric burst characteristics in a symmetric model [39], we show that entirely symmetric external input can create asymmetric burst characteristics in a symmetric model, due to the inherent asymmetry involved in anti-phasic activity. Using a pulse applied symmetrically to both neurons, we could obtain an asymmetric response characterized by an asymmetric evolution of BD and IBI for neurons #1 and #2, depending on the phase of pulse onset. This asymmetry occurred due to the anti-phasic activity of the two neurons. The same pulse was applied to both neurons, but if it was applied during the silent phase of neuron #1 and the spiking phase of neuron #2, then BD in the elicited paw-shake-like response would grow faster for neuron #2 and IBI would grow faster for neuron #1. Since neuron #2 was spiking, the slow calcium current was inactivated, i.e. its inactivation state variable h CaS2 was decaying towards 0. At the same time, the membrane potential of neuron #1 was hyperpolarized, and the slow calcium current was deinactivated, its inactivation state variable h CaS1 was growing towards 1. Thus, h CaS1 was larger than h CaS2 at the time the pulse was applied due to the asymmetric removal of inactivation of the slowest inward current. During the pulse, the slow calcium current inactivated in both neurons, i.e. the values of h CaS in both neurons were decreasing, but at the end of the pulse, h CaS was still greater for neuron #1 than for neuron #2 (Fig 6D). During transient activity for neuron #2, the value of h CaS at the beginning of the burst increased at each consecutive burst causing BD to increase throughout the transient response, and this, in turn, caused IBI of neuron #1 to increase throughout the transient response (Fig 6). During transient activity for neuron #1, the values of h CaS at the beginning of the burst made a very slight U-shape as it progressed, causing BD 1 of neuron to remain roughly constant throughout transience, which caused IBI 2 to also remain roughly constant (Fig 6). Therefore, although the current I NaS drives the stable fast rhythm, the current I CaS , driving the stable slow rhythm, controls the duration and dynamics of the transient fast activity. During the stable fast rhythm, I CaS is completely inactivated so when I CaS is only slightly deinactivated, it has a large effect on the BDs of the two neurons (Fig 5). In our model, the value of h CaS only increased to around 0.02 before the transition to the slow rhythm (Fig 7). Interestingly, these small differences in the h CaS values between neurons #1 and #2 were able to create substantial asymmetry in burst durations between the two neurons (Figs 6 and 7).
It has been theorized that slowly inactivating sodium current is the main driving current of the mammalian locomotor rhythm [20]. There is experimental evidence indicating that slowly inactivating sodium current plays an important role in the generation of fictive locomotion in the isolated rodent spinal cord [40]. Although slowly inactivating low-voltage-activated calcium current is the main driving current of the locomotor-like rhythm in our model, slowly inactivating sodium current is still required to generate locomotor-like activity. There is also experimental evidence indicating that slowly inactivating low-voltage-activated calcium current contributes to the fictive locomotor rhythm in the isolated neonatal mouse spinal cord, but it has not been determined whether this current is crucial to the generation of this rhythm [41,42].

Possible organization of a mammalian multifunctional locomotor CPG capable of producing faster transient rhythmic behaviors
In this study, we investigated a simple half-center oscillator CPG generating two phases (flexor and extensor) of a steady-state rhythmic locomotor-like behavior and a paw-shake-like transient activity. One limitation of our model is that it does not consider sensory feedback despite that motion-dependent somatosensory feedback affects the centrally generated experimental paw-shake activity patterns. However, these effects appear to be limited to two muscle groups responsible for the atypical muscle synergy during paw-shaking-the vasti (VA) and tibialis anterior (TA) muscles [37]. The phase of the VA activity burst in the cycle of real and fictive locomotion [25,[43][44][45] or real and fictive paw-shaking [19,37,38,46,47] is different from activity burst phases of other pure hindlimb extensors. This suggests a unique organization of spinal networks controlling VA motoneurons. In this study, we found that VA did not have the same BD trends as other extensors, which is consistent with the previous findings that VA's phase relationships during paw-shaking are also different from other extensors [19,25,37,38,46,47].
We have previously shown by using a neuromechanical model of cat hindlimbs that a locomotor CPG comprising a half-center oscillator rhythm generator and a basic reflex network is capable of reproducing both walking and paw-shaking [17]. In that extended model, the flexor and extensor half-centers activated flexor and extensor motoneurons, respectively, and the neuromechanical model reproduced co-activation of knee extensor vasti and ankle flexor tibialis anterior, a.k.a. the paw-shake mixed synergy [19], due to additional strong excitatory input from spindle length-sensitive afferents of vasti being stretched during the flexor phase. Thus, our previous modeling results, the reciprocal activity of hindlimb muscles with flexorand extensor-related EMG bursts in this (Fig 9) and other studies [17], as well as experimental studies of fictive paw-shake-like activity patterns generated without motion-dependent sensory input in the cat [38], have demonstrated that most of flexor and extensor hindlimb motoneuronal pools have reciprocal activity patterns consistent with receiving flexor and extensor inputs from a half-center oscillator CPG. There have been suggestions that the same or a largely shared CPG rhythm generator with flexor and extensor half-centers could generate both slow locomotor-like and fast scratch-like fictive activity patterns in the cat [48,49]. These suggestions were based on similar effects of deletions and afferent stimulations seen during fictive locomotion and scratching [48,49]. Although the duration of flexor and extensor phases are different between fictive locomotion and fictive scratching, suggesting that these two behaviors could be controlled by two specialized spinal rhythm-generating mechanisms [50], these observations appear consistent with the behavior of a single multistable HCO CPG during locomotor-like and transient paw-shake like activities (Figs 5, 6 and 9-11).
Although the exact structure of the mammalian spinal locomotor CPG is still debated, all current CPG schematics contain HCO CPG elements that provide excitatory inputs to flexor and extensor motoneurons of all muscles throughout the hindlimb [51,52]. While the more realistic model of the HCO locomotion CPG in mammals would consist of two inhibitory populations of neurons carrying mutual inhibition between two excitatory populations of neurons, we model this HCO with a single neuron representing an excitatory neural population with synaptic transmission representing signals carried by an inhibitory population. This simplification is a common practice used in computational modeling studies assuming highly synchronized activity of the two populations making up one half-center [36,53]. This simplification would miss patterns of activity with more complex relationships between neurons within and between excitatory and inhibitory populations. We suggest that our simple model represents the key temporal features of the measured EMG signals of different cat's muscles during locomotor and paw shaking rhythmic movements [36,53]. Our simplified HCO model may provide useful insights into potential mechanisms of the transition from locomotor-like rhythms to faster paw-shake like activities and their generation.
In summary, we investigated a model of a multifunctional locomotor CPG, which exhibited coexistence of a slow locomotor regime and a fast paw-shaking regime and found that transient paw-shake-like responses exhibit functionally asymmetric phase duration changes between the half-centers, and that this asymmetry depends on the phase of the locomotor-like rhythm at which the perturbation was applied. In the model, the asymmetry is caused by asymmetric levels of inactivation across the two half-centers of the slowly inactivating inward current that drives the locomotor-like rhythm. In experiments, muscle activity driving a pawshake response evoked during cat locomotion is transient in nature and consists of high-frequency bursts with increasing cycle durations and a similar asymmetry in phase duration changes. Our study makes a general prediction that multifunctionality of a CPG producing two patterns, one with a slow steady rhythm and the other with a fast transient rhythm, could potentially be identified by trends exhibited during the fast rhythm. The fast rhythm would exhibit asymmetry in its pattern imposed by the state of the shared slow variable participating in the slow rhythm.
Supporting information S1 Data. Folder contains 44 supplementary data files each corresponding to a single episode of paw shaking. File name of each file has the following structure: SupplementaryFile-Number_Data_SessionNumber_AnimalCode_EpisodeNumber.txt. Example: S1_Data_7_3-25-0_bo_9F.txt, where S1 is the supplementary data file number 7, 3-25-0 is session number, bo is animal code, 9F is episode number, and txt is file type extension. Each column of each file contains raw EMG signal of a specific muscle in mV sampled at 3000 Hz and band-pass filtered (Butterworth zero-lag filter, 30 Hz-1000 Hz, 3 dB). First line of each file contains muscle name abbreviations (see Table 3