Altered cortical synaptic lipid signaling leads to intermediate phenotypes of mental disorders.

Excitation/inhibition (E/I) balance plays important roles in mental disorders. Bioactive phospholipids like lysophosphatidic acid (LPA) are synthesized by the enzyme autotaxin (ATX) at cortical synapses and modulate glutamatergic transmission, and eventually alter E/I balance of cortical networks. Here, we analyzed functional consequences of altered E/I balance in 25 human subjects induced by genetic disruption of the synaptic lipid signaling modifier PRG-1, which were compared to 25 age and sex matched control subjects. Furthermore, we tested therapeutic options targeting ATX in a related mouse line. Using EEG combined with TMS in an instructed fear paradigm, neuropsychological analysis and an fMRI based episodic memory task, we found intermediate phenotypes of mental disorders in human carriers of a loss-of-function single nucleotide polymorphism of PRG-1 (PRG-1R345T/WT). Prg-1R346T/WT animals phenocopied human carriers showing increased anxiety, a depressive phenotype and lower stress resilience. Network analysis revealed that coherence and phase-amplitude coupling were altered by PRG-1 deficiency in memory related circuits in humans and mice alike. Brain oscillation phenotypes were restored by inhibtion of ATX in Prg-1 deficient mice indicating an interventional potential for mental disorders.

Alterations of Functional Connectivity Dynamics in Affective and Psychotic Disorders.

BACKGROUND: Patients with psychosis and patients with depression exhibit widespread neurobiological abnormalities. The analysis of dynamic functional connectivity (dFC) allows for the detection of changes in complex brain activity patterns, providing insights into common and unique processes underlying these disorders. METHODS: We report the analysis of dFC in a large sample including 127 patients at clinical high risk for psychosis, 142 patients with recent-onset psychosis, 134 patients with recent-onset depression, and 256 healthy control participants. A sliding window-based technique was used to calculate the time-dependent FC in resting-state magnetic resonance imaging data, followed by clustering to reveal recurrent FC states in each diagnostic group. RESULTS: We identified 5 unique FC states, which could be identified in all groups with high consistency (mean r = 0.889 [SD = 0.116]). Analysis of dynamic parameters of these states showed a characteristic increase in the lifetime and frequency of a weakly connected FC state in patients with recent-onset depression (p < .0005) compared with the other groups and a common increase in the lifetime of an FC state characterized by high sensorimotor and cingulo-opercular connectivities in all patient groups compared with the healthy control group (p < .0002). Canonical correlation analysis revealed a mode that exhibited significant correlations between dFC parameters and clinical variables (r = 0.617, p < .0029), which was associated with positive psychosis symptom severity and several dFC parameters. CONCLUSIONS: Our findings indicate diagnosis-specific alterations of dFC and underline the potential of dynamic analysis to characterize disorders such as depression and psychosis and clinical risk states.

Existence of a Long-Range Caudo-Rostral Sensory Influence in Terrestrial Locomotion.

In multisegmented locomotion, coordination of all appendages is crucial for the generation of a proper motor output. In running for example, leg coordination is mainly based on the central interaction of rhythm generating networks, called central pattern generators (CPGs). In slower forms of locomotion, however, sensory feedback, which originates from sensory organs that detect changes in position, velocity and load of the legs' segments, has been shown to play a more crucial role. How exactly sensory feedback influences the activity of the CPGs to establish functional neuronal connectivity is not yet fully understood. Using the female stick insect Carausius morosus, we show for the first time that a long-range caudo-rostral sensory connection exists and highlight that load as sensory signal is sufficient to entrain rhythmic motoneuron (MN) activity in the most rostral segment. So far, mainly rostro-caudal influencing pathways have been investigated where the strength of activation, expressed by the MN activity in the thoracic ganglia, decreases with the distance from the stepping leg to these ganglia. Here, we activated CPGs, producing rhythmic neuronal activity in the thoracic ganglia by using the muscarinic agonist pilocarpine and enforced the stepping of a single, remaining leg. This enabled us to study sensory influences on the CPGs' oscillatory activity. Using this approach, we show that, in contrast to the distance-dependent activation of the protractor-retractor CPGs in different thoracic ganglia, there is no such dependence for the entrainment of the rhythmic activity of active protractor-retractor CPG networks by individual stepping legs.SIGNIFICANCE STATEMENT We show for the first time that sensory information is transferred not only to the immediate adjacent segmental ganglia but also to those farther away, indicating the existence of a long-range caudo-rostral sensory influence. This influence is dependent on stepping direction but independent of whether the leg is actively or passively moved. We suggest that the sensory information comes from unspecific load signals sensed by cuticle mechanoreceptors (campaniform sensilla) of a leg. Our results provide a neuronal basis for the long-established behavioral rules of insect leg coordination. We thus provide a breakthrough in understanding the neuronal networks underlying multilegged locomotion and open new vistas into the neuronal functional connectivity of multisegmented locomotion systems across the animal kingdom.

Robustness of individualized inferences from longitudinal resting state EEG dynamics.

Tracking how individual human brains change over extended timescales is crucial to clinical scenarios ranging from stroke recovery to healthy aging. The use of resting state (RS) activity for tracking is a promising possibility. However, it is unresolved how a person's RS activity over time can be decoded to distinguish neurophysiological changes from confounding cognitive variability. Here, we develop a method to screen RS activity changes for these confounding effects by formulating it as a problem of change classification. We demonstrate a novel solution to change classification by linking individual-specific change to inter-individual differences. Individual RS-electroencephalography (EEG) was acquired over 5 consecutive days including task states devised to simulate the effects of inter-day cognitive variation. As inter-individual differences are shaped by neurophysiological differences, the inter-individual differences in RS activity on 1 day were analysed (using machine learning) to identify distinctive configurations in each individual's RS activity. Using this configuration as a decision rule, an individual could be re-identified from 2-s samples of the instantaneous oscillatory power spectrum acquired on a different day both from RS and confounded RS with a limited loss in accuracy. Importantly, the low loss in accuracy in cross-day versus same-day classification was achieved with classifiers that combined information from multiple frequency bands at channels across the scalp (with a concentration at characteristic fronto-central and occipital zones). Taken together, these findings support the technical feasibility of screening RS activity for confounding effects and the suitability of longitudinal RS for robust individualized inferences about neurophysiological change in health and disease.

Stimulus transformation into motor action: Dynamic graph analysis reveals a posterior-to-anterior shift in brain network communication of older subjects.

Cognitive performance slows down with increasing age. This includes cognitive processes that are essential for the performance of a motor act, such as the slowing down in response to an external stimulus. The objective of this study was to identify aging-associated functional changes in the brain networks that are involved in the transformation of external stimuli into motor action. To investigate this topic, we employed dynamic graphs based on phase-locking of Electroencephalography signals recorded from healthy younger and older subjects while performing a simple visually-cued finger-tapping task. The network analysis yielded specific age-related network structures varying in time in the low frequencies (2-7 Hz), which are closely connected to stimulus processing, movement initiation and execution in both age groups. The networks in older subjects, however, contained several additional, particularly interhemispheric, connections and showed an overall increased coupling density. Cluster analyses revealed reduced variability of the subnetworks in older subjects, particularly during movement preparation. In younger subjects, occipital, parietal, sensorimotor and central regions were-temporally arranged in this order-heavily involved in hub nodes. Whereas in older subjects, a hub in frontal regions preceded the noticeably delayed occurrence of sensorimotor hubs, indicating different neural information processing in older subjects. All observed changes in brain network organization, which are based on neural synchronization in the low frequencies, provide a possible neural mechanism underlying previous fMRI data, which report an overactivation, especially in the prefrontal and pre-motor areas, associated with a loss of hemispheric lateralization in older subjects.

The roles of ascending sensory signals and top-down central control in the entrainment of a locomotor CPG.

Previous authors have proposed two basic hypotheses about the factors that form the basis of locomotor rhythms in walking insects: sensory feedback only or sensory feedback together with rhythmic activity of small neural circuits called central pattern generators (CPGs). Here we focus on the latter. Following this concept, to generate functional outputs, locomotor control must feature both rhythm generation by CPGs at the level of individual joints and coordination of their rhythmic activities, so that all muscles are activated in an appropriate pattern. This work provides an in-depth analysis of an aspect of this coordination process based on an existing network model of stick insect locomotion. Specifically, we consider how the control system for a single joint in the stick insect leg may produce rhythmic output when subjected to ascending sensory signals from other joints in the leg. In this work, the core rhythm generating CPG component of the joint under study is represented by a classical half-center oscillator constrained by a basic set of experimental observations. While the dynamical features of this CPG, including phase transitions by escape and release, are well understood, we provide novel insights about how these transition mechanisms yield entrainment to the incoming sensory signal, how entrainment can be lost under variation of signal strength and period or other perturbations, how entrainment can be restored by modulation of tonic top-down drive levels, and how these factors impact the duty cycle of the motor output.

A response-locking protocol to boost sensitivity for fMRI-based neurochronometry.

The timeline of brain-wide neural activity relative to a behavioral event is crucial when decoding the neural implementation of a cognitive process. Yet, fMRI assesses neural processes indirectly via delayed and regionally variable hemodynamics. This method-inherent temporal distortion impacts the interpretation of behavior-linked neural timing. Here we describe a novel behavioral protocol that aims at disentangling the BOLD dynamics of the pre- and post-response periods in response time tasks. We tested this response-locking protocol in a perceptual decision-making (random dot) task. Increasing perceptual difficulty produced expected activity increases over a broad network involving the lateral/medial prefrontal cortex and the anterior insula. However, response-locking revealed a previously unreported functional dissociation within this network. preSMA and anterior premotor cortex (prePMV) showed post-response activity modulations while anterior insula and anterior cingulate cortex did not. Furthermore, post-response BOLD activity at preSMA showed a modulation in timing but not amplitude while this pattern was reversed at prePMV. These timeline dissociations with response-locking thus revealed three functionally distinct sub-networks in what was seemingly one shared distributed network modulated by perceptual difficulty. These findings suggest that our novel response-locked protocol could boost the timing-related sensitivity of fMRI.

Unravelling intra- and intersegmental neuronal connectivity between central pattern generating networks in a multi-legged locomotor system.

Animal walking results from a complex interplay of central pattern generating networks (CPGs), local sensory signals expressing position, velocity and forces generated in the legs, and coordinating signals between neighboring legs. In particular, the CPGs control the activity of motoneuron (MN) pools which drive the muscles of the individual legs and are thereby responsible for the generation of rhythmic leg movements. The rhythmic activity of the CPGs as well as their connectivity can be modified by the aforementioned sensory signals. However, the precise nature of the interaction between the CPGs and these sensory signals has remained generally largely unknown. Experimental methods aiming at finding out details of these interactions often apply cholinergic agonists such as pilocarpine in order to induce rhythmic activity in the CPGs. Using this general approach, we removed the influence of sensory signals and investigated the putative connections between CPGs controlling the upward/downward movement in the different legs of the stick insect. The experimental data, i.e. the measured MN activities, underwent connectivity analysis using Dynamic Causal Modelling (DCM). This method can uncover the underlying coupling structure and strength between pairs of segmental CPGs. For the analysis we set up different coupling schemes (models) for DCM and compared them using Bayesian Model Selection (BMS). Models with contralateral connections in each segment and ipsilateral connections on both sides, as well as the coupling from the meta- to the ipsilateral prothoracic ganglion were preferred by BMS to all other types of models tested. Moreover, the intrasegmental coupling strength in the mesothoracic ganglion was the strongest and most stable in all three ganglia.

A kinematic model of stick-insect walking.

Animal, and insect walking (locomotion) in particular, have attracted much attention from scientists over many years up to now. The investigations included behavioral, electrophysiological experiments, as well as modeling studies. Despite the large amount of material collected, there are left many unanswered questions as to how walking and related activities are generated, maintained, and controlled. It is obvious that for them to take place, precise coordination within muscle groups of one leg and between the legs is required: intra- and interleg coordination. The nature, the details, and the interactions of these coordination mechanisms are not entirely clear. To help uncover them, we made use of modeling techniques, and succeeded in developing a six-leg model of stick-insect walking. Our main goal was to prove that the same model can mimic a variety of walking-related behavioral modes, as well as the most common coordination patterns of walking just by changing the values of a few input or internal variables. As a result, the model can reproduce the basic coordination patterns of walking: tetrapod and tripod and the transition between them. It can also mimic stop and restart, change from forward-to-backward walking and back. Finally, it can exhibit so-called search movements of the front legs both while walking or standing still. The mechanisms of the model that enable it to produce the aforementioned behavioral modes can hint at and prove helpful in uncovering further details of the biological mechanisms underlying walking.

Reconstruction of effective connectivity in the case of asymmetric phase distributions.

BACKGROUND: The interaction of different brain regions is supported by transient synchronization between neural oscillations at different frequencies. Different measures based on synchronization theory are used to assess the strength of the interactions from experimental data. One method of estimating the effective connectivity between brain regions, within the framework of the theory of weakly coupled phase oscillators, was implemented in Dynamic Causal Modelling (DCM) for phase coupling (Penny et al., 2009). However, the results of such an approach strongly depend on the observables used to reconstruct the equations (Kralemann et al., 2008). In particular, an asymmetric distribution of the observables could result in a false estimation of the effective connectivity between the network nodes. NEW METHOD: In this work we built a new modelling part into DCM for phase coupling, and extended it with a distortion function that accommodates departures from purely sinusoidal oscillations. RESULTS: By analysing numerically generated data sets with an asymmetric phase distribution, we demonstrated that the extended DCM for phase coupling with the additional modelling component, correctly estimates the coupling functions. COMPARISON WITH EXISTING METHODS: The new method allows for different intrinsic frequencies among coupled neuronal populations and provides results that do not depend on the distribution of the observables. CONCLUSIONS: The proposed method can be used to analyse effective connectivity between brain regions within and between different frequency bands, to characterize m:n phase coupling, and to unravel underlying mechanisms of the transient synchronization.

Freely chosen and instructed actions are terminated by different neural mechanisms revealed by kinematics-informed EEG.

Neurophysiological accounts of human volition are dominated by debates on the origin of voluntary choices but the neural consequences that follow such choices remain poorly understood. For instance, could one predict whether or not an action was chosen voluntarily based only on how that action is motorically executed? We investigated this possibility by integrating scalp electroencephalograms and index-finger accelerometer recordings acquired while people chose between pressing a left or right button either freely or as instructed by a visual cue. Even though freely selected and instructed actions were executed with equal vigor, the timing of the movement to release the button was comparatively delayed for freely selected actions. This chronometric difference was six-times larger for the ß-oscillations over the sensorimotor cortex that characteristically accompany an action's termination. This surprising modulation of an action's termination by volition was traceable to volition-modulated differences in how the competing yet non-selected action was represented and regulated.

Aging-associated changes of movement-related functional connectivity in the human brain.

Motor performance declines with normal aging. Previous neuroimaging work revealed aging-related general increases in neural activity, especially in the prefrontal and pre-motor areas, associated with a loss of hemispheric lateralization. However, the functional mechanisms underlying these changes and their relation to aging-associated motor decline to date remain elusive. To further elucidate the neural processes underlying aging-related motor decline, we recorded EEG from younger and older subjects while they performed a finger-tapping task. As a measure of synchronization between motor areas, we computed the inter-regional phase-locking value which reflects functional connectivity between distinct neural populations. Behavioral data revealed increased movement times in older subjects. EEG data showed that phase locking in the δ-θ frequencies is a general, age-independent phenomenon underlying the execution of simple finger movements. In stark contrast, the extent of synchronization between motor areas significantly differed dependent upon age of subjects: multiple additional intra- and inter-hemispheric connections were observed in older subjects. Our data shed light upon the results of previous neuroimaging studies showing aging-related increases in neural activation. In particular, data suggest that the observed aging-dependent substantial intra- and inter-hemispheric reorganization of connectivity between the corresponding motor areas underlies the previously reported loss of lateralization in older subjects. The changes observed are likely to represent compensatory mechanisms aiming at preserved task performance in older subjects.

The role of phase shifts of sensory inputs in walking revealed by means of phase reduction.

Detailed neural network models of animal locomotion are important means to understand the underlying mechanisms that control the coordinated movement of individual limbs. Daun-Gruhn and Tóth, Journal of Computational Neuroscience 31(2), 43-60 (2011) constructed an inter-segmental network model of stick insect locomotion consisting of three interconnected central pattern generators (CPGs) that are associated with the protraction-retraction movements of the front, middle and hind leg. This model could reproduce the basic locomotion coordination patterns, such as tri- and tetrapod, and the transitions between them. However, the analysis of such a system is a formidable task because of its large number of variables and parameters. In this study, we employed phase reduction and averaging theory to this large network model in order to reduce it to a system of coupled phase oscillators. This enabled us to analyze the complex behavior of the system in a reduced parameter space. In this paper, we show that the reduced model reproduces the results of the original model. By analyzing the interaction of just two coupled phase oscillators, we found that the neighboring CPGs could operate within distinct regimes, depending on the phase shift between the sensory inputs from the extremities and the phases of the individual CPGs. We demonstrate that this dependence is essential to produce different coordination patterns and the transition between them. Additionally, applying averaging theory to the system of all three phase oscillators, we calculate the stable fixed points - they correspond to stable tripod or tetrapod coordination patterns and identify two ways of transition between them.

Age-related changes in oscillatory power affect motor action.

With increasing age cognitive performance slows down. This includes cognitive processes essential for motor performance. Additionally, performance of motor tasks becomes less accurate. The objective of the present study was to identify general neural correlates underlying age-related behavioral slowing and the reduction in motor task accuracy. To this end, we continuously recorded EEG activity from 18 younger and 24 older right-handed healthy participants while they were performing a simple finger tapping task. We analyzed the EEG records with respect to local changes in amplitude (power spectrum) as well as phase locking between the two age groups. We found differences between younger and older subjects in the amplitude of post-movement synchronization in the ß band of the sensory-motor and medial prefrontal cortex (mPFC). This post-movement ß amplitude was significantly reduced in older subjects. Moreover, it positively correlated with the accuracy with which subjects performed the motor task at the electrode FCz, which detects activity of the mPFC and the supplementary motor area. In contrast, we found no correlation between the accurate timing of local neural activity, i.e. phase locking in the δ-θ frequency band, with the reaction and movement time or the accuracy with which the motor task was performed. Our results show that only post-movement ß amplitude and not δ-θ phase locking is involved in the control of movement accuracy. The decreased post-movement ß amplitude in the mPFC of older subjects hints at an impaired deactivation of this area, which may affect the cognitive control of stimulus-induced motor tasks and thereby motor output.

Modeling search movements of an insect's front leg.

Beside locomotion, search movements are another important type of motor activity of insects. They are very often performed by the front legs of the animals. They consist of cyclic stereotypical leg movements that can be modified by sensory signals. The details of the local organization of these movements have however not yet been studied. In this paper, we, using an appropriate variant of our existing one-leg model, present a scheme of how these searching movements might be organized and performed on the level of local neuromuscular control networks. In the simulations with the model, we attempted to mimic the experimental results by Berg et al. (J. Exp. Biol. 216:1064-1074, 2013) in which an obstacle was put in the way of the search movements of the front leg for a very short while, and then the recovery to the usual search movements was observed and analyzed. Our simulation results suggest that the recruitment of the fast levator and depressor muscles play a crucial part in resuming the search movements after removal of the obstacle. The interplay between the levator and depressor, and the extensor and flexor local control networks can, according to the model, bring about a large variety of search movements upon removal of the obstacle. A number of these movements are comparable with those seen in the experiments.

Frequency-specific modulation of connectivity in the ipsilateral sensorimotor cortex by different forms of movement initiation.

A consistent finding in motor EEG research is a bilateral attenuation of oscillatory activity over sensorimotor regions close to the onset of an upcoming unilateral hand movement. In contrast, little is known about how movement initiation affects oscillatory activity, especially in the hemisphere ipsilateral to the moving hand. We here investigated the neural mechanisms modulating oscillatory activity in the ipsilateral motor cortex prior to movement onset under the control of two different initiating networks, namely, Self-initiated and Visually-cued actions. During motor preparation, a contralateral preponderance of power over sensorimotor cortex (SM) was observed in α and ß bands during Visually-cued movements, whereas power changes were more bilateral during Self-initiated movements. Coherence between ipsilateral SM (iSM) and contralateral SM (cSM) in the α-band was significantly increased compared to the respective baseline values, independent of the context of movement initiation. However, this context-independent cSM-iSM coherence modulated the power changes in iSM in a context-dependent manner, that is, a stronger cSM-iSM coherence correlated with a larger decrease in high-ß power over iSM in the Self-initiated condition, in contrast to a smaller decrease in α power in the Visually-cued condition. In addition, the context-dependent coherence between SMA and iSM in the α-band and δ-θ-band for the Self-initiated and Visually-cued condition, respectively, exhibited a similar context-dependent modulation for power changes. Our findings suggest that the initiation of regional oscillations over iSM reflects changes in the information flow with the contralateral sensorimotor and premotor areas dependent upon the context of movement initiation. Importantly, the interaction between regional oscillations and network-like oscillatory couplings indicates different frequency-specific inhibitory mechanisms that modulate the activity in the ipsilateral sensorimotor cortex dependent upon how the movement is initiated.

Intra- and intersegmental influences among central pattern generating networks in the walking system of the stick insect.

To efficiently move around, animals need to coordinate their limbs. Proper, context-dependent coupling among the neural networks underlying leg movement is necessary for generating intersegmental coordination. In the slow-walking stick insect, local sensory information is very important for shaping coordination. However, central coupling mechanisms among segmental central pattern generators (CPGs) may also contribute to this. Here, we analyzed the interactions between contralateral networks that drive the depressor trochanteris muscle of the legs in both isolated and interconnected deafferented thoracic ganglia of the stick insect on application of pilocarpine, a muscarinic acetylcholine receptor agonist. Our results show that depressor CPG activity is only weakly coupled between all segments. Intrasegmental phase relationships differ between the three isolated ganglia, and they are modified and stabilized when ganglia are interconnected. However, the coordination patterns that emerge do not resemble those observed during walking. Our findings are in line with recent studies and highlight the influence of sensory input on coordination in slowly walking insects. Finally, as a direct interaction between depressor CPG networks and contralateral motoneurons could not be observed, we hypothesize that coupling is based on interactions at the level of CPG interneurons.NEW & NOTEWORTHY Maintaining functional interleg coordination is vitally important as animals locomote through changing environments. The relative importance of central mechanisms vs. sensory feedback in this process is not well understood. We analyzed coordination among the neural networks generating leg movements in stick insect preparations lacking phasic sensory feedback. Under these conditions, the networks governing different legs were only weakly coupled. In stick insect, central connections alone are thus insufficient to produce the leg coordination observed behaviorally.

Effects of functional decoupling of a leg in a model of stick insect walking incorporating three ipsilateral legs.

Legged locomotion is a fundamental form of activity of insects during which the legs perform coordinated movements. Sensory signals conveying position, velocity and load of a leg are sent between the thoracic ganglia where the local control networks of the leg muscles are situated. They affect the actual state of the local control networks, hence the stepping of the legs. Sensory coordination in stepping has been intensively studied but important details of its neuronal mechanisms are still unclear. One possibility to tackle this problem is to study what happens to the coordination if a leg is, reversibly or irreversibly, deprived of its normal function. There are numerous behavioral studies on this topic but they could not fully uncover the underlying neuronal mechanisms. Another promising approach to make further progress here can be the use of appropriate models that help elucidate those coordinating mechanisms. We constructed a model of three ipsilateral legs of a stick insect that can mimic coordinated stepping of these legs. We used this model to investigate the possible effects of decoupling a leg. We found that decoupling of the front or the hind leg did not disrupt the coordinated walking of the two remaining legs. However, decoupling of the middle leg yielded mixed results. Both disruption and continuation of coordinated stepping of the front and hind leg occurred. These results agree with the majority of corresponding experimental findings. The model suggests a number of possible mechanisms of decoupling that might bring about the changes.

Control of oscillation periods and phase durations in half-center central pattern generators: a comparative mechanistic analysis.

Central pattern generators (CPGs) consisting of interacting groups of neurons drive a variety of repetitive, rhythmic behaviors in invertebrates and vertebrates, such as arise in locomotion, respiration, mastication, scratching, and so on. These CPGs are able to generate rhythmic activity in the absence of afferent feedback or rhythmic inputs. However, functionally relevant CPGs must adaptively respond to changing demands, manifested as changes in oscillation period or in relative phase durations in response to variations in non-patterned inputs or drives. Although many half-center CPG models, composed of symmetric units linked by reciprocal inhibition yet varying in their intrinsic cellular properties, have been proposed, the precise oscillatory mechanisms operating in most biological CPGs remain unknown. Using numerical simulations and phase-plane analysis, we comparatively investigated how the intrinsic cellular features incorporated in different CPG models, such as subthreshold activation based on a slowly inactivating persistent sodium current, adaptation based on slowly activating calcium-dependent potassium current, or post-inhibitory rebound excitation, can contribute to the control of oscillation period and phase durations in response to changes in excitatory external drive to one or both half-centers. Our analysis shows that both the sensitivity of oscillation period to alterations of excitatory drive and the degree to which the duration of each phase can be separately controlled depend strongly on the intrinsic cellular mechanisms involved in rhythm generation and phase transitions. In particular, the CPG formed from units incorporating a slowly inactivating persistent sodium current shows the greatest range of oscillation periods and the greatest degree of independence in phase duration control by asymmetric inputs. These results are explained based on geometric analysis of the phase plane structures corresponding to the dynamics for each CPG type, which in particular helps pinpoint the roles of escape and release from synaptic inhibition in the effects we find.

Equation-based models of dynamic biological systems.

The purpose of this review is to introduce differential equations as a simulation tool in the biological and clinical sciences. This modeling technique is very mature and has been a preferred tool of physiologists and bioengineers and of quantitative scientists in general to describe and predict the behavior of complex interacting systems. However, this methodology has not been widely used within clinical medicine due to a lack of familiarity with highly quantitative methods and a greater acquaintance with statistical modeling approaches based on inference and empirical data analysis. We will describe various aspects of equation-based modeling, including underlying assumptions, strengths, and weaknesses and provide specific examples of simple models. We conclude that the usefulness of quantitative modeling, including equation-based models, is ultimately linked to the quality and abundance of observation obtained on the system being modeled. Equation-based modeling, although potentially an integrative approach, is complementary to and extends the potential of traditional statistically based approaches to inference.