Stochastic versus deterministic variability in simple neuronal circuits: II. Hippocampal slice.

Long time series of Schaffer collateral to CA1 pyramidal cell presynaptic volleys (stratum radiatum) and population spikes (stratum pyramidale) were evoked (driven) in rat hippocampal slices. From the driven CA1 region in normal [K+] perfusate, both population spike amplitude and an input-output function consisting of population spike amplitude divided by the presynaptic volley amplitude were analyzed. Raising [K+] in the perfusion medium to 8.5 mM, slices were induced to spontaneously burst fire in CA3 and long time series of inter-burst intervals were recorded. Three tests for determinism were applied to these series: a discrete adaptation of a local flow approach, a local dispersion approach, and nonlinear prediction. Surrogate data were generated to serve as mathematical and statistical controls. All of the population spike (6/6) and input-output (6/6) time series from the normal [K+] driven circuitry were stochastic by all three methods. Although most of the time series (5/6) from the autonomously bursting high [K+] state failed to demonstrate evidence of determinism, one (1/6) of these time series did demonstrate significant determinism. This single instance of predictability could not be accounted for by the linear correlation in these data.     

Detecting determinism in short time series, with an application to the analysis of a stationary EEG recording

We have developed a new method for detecting determinism in a short time series and used this method to examine whether a stationary EEG is deterministic or stochastic. The method is based on the observation that the trajectory of a time series generated from a differentiable dynamical system behaves smoothly in an embedded phase space. The angles between two successive directional vectors in the trajectory reconstructed from a time series at a minimum embedding dimension were calculated as a function of time. We measured the irregularity of the angle variations obtained from the time series using second-order difference plots and central tendency measures, and compared these values with those from surrogate data. The ability of the proposed method to distinguish between chaotic and stochastic dynamics is demonstrated through a number of simulated time series, including data from Lorenz, R?ssler, and Van der Pol attractors, high-dimensional equations, and 1/f noise. We then applied this method to the analysis of stationary segments of EEG recordings consisting of 750 data points (6-s segments) from five normal subjects. The stationary EEG segments were not found to exhibit deterministic components. This method can be used to analyze determinism in short time series, such as those from physiological recordings, that can be modeled using differentiable dynamical processes.     

Capsaicin-sensitive muscle afferents modulate the monosynaptic reflex in response to muscle ischemia and fatigue in the rat

The role of muscle ischemia and fatigue in modulating the monosynaptic reflex was investigated in decerebrate and spinalized rats. Field potentials and fast motoneuron single units in the lateral gastrocnemious (LG) motor pool were evoked by dorsal root stimulation. Muscle ischemia was induced by occluding the LG vascular supply and muscle fatigue by prolonged tetanic electrical stimulation of the LG motor nerve. Under muscle ischemia the monosynaptic reflex was facilitated since the size of the early and late waves of the field potential and the excitability of the motoneuron units increased. This effect was abolished after L3-L6 dorsal rhizotomy, but it was unaffected after L3-L6 ventral rhizotomy. By contrast, the monosynaptic reflex was inhibited by muscle fatiguing stimulation, and this effect did not fully depend on the integrity of the dorsal root. However, when ischemia was combined with repetitive tetanic muscle stimulation the inhibitory effect of fatigue was significantly enhanced. Both the ischemia and fatigue effects were abolished by capsaicin injected into the LG muscle at a dose that blocked a large number of group III and IV muscle afferents. We concluded that muscle ischemia and fatigue activate different groups of muscle afferents that are both sensitive to capsaicin, but enter the spinal cord through different roots. They are responsible for opposite effects, when given separately: facilitation during ischemia and inhibition during fatigue; however, in combination, ischemia enhances the responsiveness of the afferent fibres to fatigue.     

Effects of chronic electrical stimulation on paralyzed expiratory muscles.

Following spinal cord injury, the expiratory muscles develop significant disuse atrophy characterized by reductions in their weight, fiber cross-sectional area, and force-generating capacity. We determined the extent to which these physiological alterations can be prevented with electrical stimulation. Because a critical function of the expiratory muscles is cough generation, an important goal was the maintenance of maximal force production. In a cat model of spinal cord injury, short periods of high-frequency lower thoracic electrical spinal cord stimulation (SCS) at the T(10) level (50 Hz, 15 min, twice/day, 5 days/wk) were initiated 2 wk following spinalization and continued for a 6-mo period. Airway pressure (P)-generating capacity was determined by SCS. Five acute, spinalized animals served as controls. Compared with controls, initial P fell from 43.9 +/- 1.0 to 41.8 +/- 0.7 cmH(2)O (not significant) in the chronic animals. There were small reductions in the weight of the external oblique, internal oblique, transverses abdominis, internal intercostal, and rectus abdominis muscles (not significant for each). There were no significant changes in the population of fast muscle fibers. Because prior studies (Kowalski KE, Romaniuk JR, DiMarco AF. J Appl Physiol 102: 1422-1428, 2007) have demonstrated significant atrophy following spinalization in this model, these results indicate that expiratory muscle atrophy can be prevented by the application of short periods of daily high-frequency stimulation. Because the frequency of stimulation is similar to the expected pattern of clinical use for cough generation, the daily application of electrical stimulation could potentially serve the dual purpose of maintenance of expiratory muscle function and airway clearance.     

Changes in expiratory muscle function following spinal cord section.

Following spinal cord injury, muscles below the level of injury develop variable degrees of disuse atrophy. The present study assessed the physiological changes of the expiratory muscles in a cat model of spinal cord injury. Muscle fiber typing, cross-sectional area, muscle weight, and changes in pressure-generating capacity were assessed in five cats spinalized at the T(6) level. Airway pressure (P)-generating capacity was monitored during lower thoracic spinal cord stimulation before and 6 mo after spinalization. These parameters were also assessed in five acute animals, which served as controls. In spinalized animals, P fell from 41 +/- l to 28 +/- 3 cm H2O (means +/- SE; P < 0.001). Muscle weight of the external oblique, internal oblique, transversus abdominis, and internal intercostal muscles decreased significantly (P < 0.05 for each). Muscle weight of the external oblique, internal oblique, transversus abdominis, and internal intercostal, but not rectus abdominis (RA), correlated linearly with P (r > 0.7 for each; P < 0.05 for each). Mean muscle fiber cross-sectional area of these muscles was significantly smaller (P < 0.05 for each; except RA) and also correlated linearly with P (r > 0.55 for each; P < 0.05 for each, except RA). In spinalized animals, the expiratory muscles demonstrated a significant increase in the population of fast muscle fibers. These results indicate that, following spinalization, 1) the expiratory muscles undergo significant atrophy and fiber-type transformation and 2) the P-generating capacity of the expiratory muscles falls significantly secondary to reductions in muscle mass.     

Establishing the stochastic nature of intracellular calcium oscillations from experimental data

Calcium has been established as a key messenger in both intra- and intercellular signaling. Experimentally observed intracellular calcium responses to different agonists show a variety of behaviors from simple spiking to complex oscillatory regimes. Here we study typical experimental traces of calcium oscillations in hepatocytes obtained in response to phenylephrine and ATP. The traces were analyzed with methods of nonlinear time series analysis in order to determine the stochastic/deterministic nature of the intracellular calcium oscillations. Despite the fact that the oscillations appear, visually, to be deterministic yet perturbed by noise, our analyses provide strong evidence that the measured calcium traces in hepatocytes are prevalently of stochastic nature. In particular, bursting calcium oscillations are temporally correlated Gaussian series distorted by a monotonic, instantaneous, time-independent function, whilst the spiking behavior appears to have a dynamical nonlinear component whereby the overall determinism level is still low. The biological importance of this finding is discussed in relation to the mechanisms incorporated in mathematical models as well as the role of stochasticity and determinism at cellular and tissue levels which resemble typical statistical and thermodynamic effects in physics.     

Effects of peripheral inputs from hindlimb on the monosynaptic reflex of motoneurons innervating tail muscles.

The effects of group II muscle (PBSt, GS) and cutaneous afferent (Sur, SPc, Tib) inputs from the hindlimb on the monosynaptic reflexes of motoneurons innervating tail muscles were studied in lower spinalized cats. Stimulation of the cutaneous nerves at the conditioning-test stimulus interval of about 10-20 ms facilitated and inhibited the monosynaptic reflexes of ipsilateral and contralateral tail muscles, respectively. The effects of the muscle nerve stimulation were not so prominent as those elicited by cutaneous nerve stimulation. The monosynaptic reflex was also inhibited by muscle nerve stimulation at 10-50 ms intervals. The effects of conditioning stimulation of the hindlimb peripheral nerves at short intervals were depressed or blocked by section of the ipsilateral lateral funiculus at S1 spinal segment. These findings show that the neuronal pathway from hindlimb afferents to tail muscle motoneurons passed the lateral funiculus of the spinal cord and modulates the motoneuronal activity of tail muscles.     

The effects of cold-induced muscle spindle secondary activity on monosynaptic and stretch reflexes in the decerebrate cat.

The effects of muscle spindle secondary ending activity on the stretch reflex were studied in unanesthetized decerebrate cats. Activation of secondary endings was accomplished by reducing the muscle temperature. This has been shown to cause a sustained asynchronous discharge from secondary endings. Cooling of the medial gastrocnemius or lateral gastrocnemius-soleus muscles caused an increase in the phasic and tonic components of their stretch reflexes. Cooling of the relaxed medial gastrocnemius muscle caused similar increases in the components of the stretch reflex of the synergistic lateral gastrocnemius-soleus muscle and an increase in its monosynaptic reflex. It was concluded that the facilitatory autogenetic and synergistic effects of muscle cooling on the stretch and monosynaptic reflexes were brought about by activity in group II afferents from muscle spindle secondary endings and could not be ascribed to any other type of muscle receptor. These results support the concept of an excitatory role for the secondary endings of the muscle spindle in the stretch reflex of the decerebrate cat.     

Limb movements generated by stimulating muscle,nerve and spinal cord

We have compared the movements generated by stimulation of muscle, nerve, spinal roots and spinal cord in anesthetized, decerebrate and spinalized cats. Each method produced a full range of movements of the cat's hind limb in the sagittal plane against a spring load, except for stimulation of the roots. Stimulation of the dorsal roots produced movements that were mainly up and forward, whereas stimulation of the ventral roots produced complementary movements (down and backward). Results from stimulation in the intermediate areas of the spinal cord were compared to predictions of the "movement primitives" hypothesis. We could not confirm that the directions were independent of stimulus amplitude or the state of descending inputs. Pros and cons of stimulating at some sites were provisionally considered for the reliable control of limb movements with functional electrical stimulation (FES) in clinical conditions.     

Random walks, random sequences, and nonlinear dynamics in human optokinetic nystagmus.

Optokinetic nystagmus (OKN) is a reflexive eye movement with target-following slow phases (SP) alternating with oppositely directed fast phases (FP). We measured the following from OKN in three humans: FP beginning and ending positions, amplitudes, and intervals and SP amplitudes and velocities. We sought to predict future values of each parameter on the basis of past values, using state-space representation of the sequence (time-delay embedding) and local second-order approximation of trajectories. Predictability is an indication of determinism: this approach allows us to investigate the relative contributions of random and deterministic dynamics in OKN. FP beginning and ending positions showed good predictability, but SP velocity was less predictable. FP and SP amplitudes and FP intervals had little or no predictability. FP beginnings and endings were as predictable as randomized versions that retain linear autocorrelation; this is typical of random walks. Predictability of FP intervals did not change under random rearrangement, which is characteristic of a random process. Only linear determinism was demonstrated; nonlinear interactions may exist that would not be detected by our present approach.     

On the roles of time, space and habitat in a boreal small mammal assemblage: predictably stochastic assembly

Ecologists continue to debate the roles of deterministic versus stochastic (or neutral) processes in the assembly of ecological communities. The debate often hinges on issues of temporal and spatial scale. Resolution of the competing views depends on a detailed understanding of variation in the structure of local communities through time and space. Analyses of twelve years of data on a diverse assemblage of 13 boreal small mammal species revealed both deterministic and stochastic patterns. Stochastic membership in the overall community created unique assemblages of species in both time and space. But the relative abundances of the two codominant species were much less variable, and suggest a significant role for strong interactions that create temporal and spatial autocorrelation in abundance. As species wax and wane in abundance, they are nevertheless subject to probabilistic rules on local assembly. At the scales I report on here, poorly understood large scale processes influence the presence and absence of the majority of (sparse) species in the assembly. But the overall pool of species nevertheless obeys local rules on their ultimate stochastic assembly into groups of interacting species.     

Responses of nucleus z neurons to vibration of hindlimb extensor muscles in the decerebrate cat.

1. Small sinusoidal changes of length, applied longitudinally to the ddifferented GS muscle, were uused as a specific stimulus to activate the muscle spindle receptors in precollicular decerebrate cats. In order to determine the relative contribution of the primary and secondary endings of muscle spindles to the response of the nucleus z neurons, the effects of muscle vibration on this unit activity were studied under conditions in which the segmental monosynaptic reflexes produced by stimulation of the primary endings of muscle spindles were continuously monitored. 2. Vibration of the GS muscle (at 180-200/sec and amplitudes up to 250-320 mu) affected the frequency and pattern of discharge in 50 out of 168 units recorded from the lower medulla...     

Dynamics of On-Off Neural Firing Patterns and Stochastic Effects near a Sub-Critical Hopf Bifurcation

On-off firing patterns, in which repetition of clusters of spikes are interspersed with epochs of subthreshold oscillations or quiescent states, have been observed in various nervous systems, but the dynamics of this event remain unclear. Here, we report that on-off firing patterns observed in three experimental models (rat sciatic nerve subject to chronic constrictive injury, rat CA1 pyramidal neuron, and rabbit blood pressure baroreceptor) appeared as an alternation between quiescent state and burst containing multiple period-1 spikes over time. Burst and quiescent state had various durations. The interspike interval (ISI) series of on-off firing pattern was suggested as stochastic using nonlinear prediction and autocorrelation function. The resting state was changed to a period-1 firing pattern via on-off firing pattern as the potassium concentration, static pressure, or depolarization current was changed. During the changing process, the burst duration of on-off firing pattern increased and the duration of the quiescent state decreased. Bistability of a limit cycle corresponding to period-1 firing and a focus corresponding to resting state was simulated near a sub-critical Hopf bifurcation point in the deterministic Morris—Lecar (ML) model. In the stochastic ML model, noise-induced transitions between the coexisting regimes formed an on-off firing pattern, which closely matched that observed in the experiment. In addition, noise-induced exponential change in the escape rate from the focus, and noise-induced coherence resonance were identified. The distinctions between the on-off firing pattern and stochastic firing patterns generated near three other types of bifurcations of equilibrium points, as well as other viewpoints on the dynamics of on-off firing pattern, are discussed. The results not only identify the on-off firing pattern as noise-induced stochastic firing pattern near a sub-critical Hopf bifurcation point, but also offer practical indicators to discriminate bifurcation types and neural excitability types.     

Spinal cord injury-induced changes in breathing are not due to supraspinal plasticity in turtles (Pseudemys scripta)

After occurrence of spinal cord injury, it is not known whether the respiratory rhythm generator undergoes plasticity to compensate for respiratory insufficiency. To test this hypothesis, respiratory variables were measured in adult semiaquatic turtles using a pneumotachograph attached to a breathing chamber on a water-filled tank. Turtles breathed room air (2 h) before being challenged with two consecutive 2-h bouts of hypercapnia (2 and 6% CO2 or 4 and 8% CO2). Turtles were spinalized at dorsal segments D8-D10 so that only pectoral girdle movement was used for breathing. Measurements were repeated at 4 and 8 wk postinjury. For turtles breathing room air, breathing frequency, tidal volume, and ventilation were not altered by spinalization; single-breath (singlet) frequency increased sevenfold. Spinalized turtles breathing 6-8% CO2 had lower ventilation due to decreased frequency and tidal volume, episodic breathing (breaths/episode) was reduced, and singlet breathing was increased sevenfold. Respiratory variables in sham-operated turtles were unaltered by surgery. Isolated brain stems from control, spinalized, and sham turtles produced similar respiratory motor output and responded the same to increased bath pH. Thus spinalized turtles compensated for pelvic girdle loss while breathing room air but were unable to compensate during hypercapnic challenges. Because isolated brain stems from control and spinalized turtles had similar respiratory motor output and chemosensitivity, breathing changes in spinalized turtles in vivo were probably not due to plasticity within the respiratory rhythm generator. Instead, caudal spinal cord damage probably disrupts spinobulbar pathways that are necessary for normal breathing.     

Movements generated by intraspinal microstimulation in the intermediate gray matter of the anesthetized, decerebrate, and spinal cat

The intermediate laminae of the lumbosacral spinal cord are suggested to contain a small number of specialized neuronal circuits that form the basic elements of movement construction ("movement primitives"). Our aim was to study the properties and state dependence of these hypothesized circuits in comparison with movements elicited by direct nerve or muscle stimulation. Microwires for intraspinal microstimulation (ISMS) were implanted in intermediate laminae throughout the lumbosacral enlargement. Movement vectors evoked by ISMS were compared with those evoked by stimulation through muscle and nerve electrodes in cats that were anesthetized, then decerebrated, and finally spinalized. Similar movements could be evoked under anesthesia by ISMS and nerve and muscle stimulation, and these covered the full work space of the limb. ISMS-evoked movements were associated with the actions of nearby motoneuron pools. However, after decerebration and spinalization, ISMS-evoked movements were dominated by flexion, with few extensor movements. This indicates that the outputs of neuronal networks in the intermediate laminae depend significantly on descending input and on the state of the spinal cord. Frequently, the outputs also depended on stimulus intensity. These experiments suggest that interneuronal circuits in the intermediate and ventral regions of the spinal cord overlap and their function may be to process reflex and descending activity in a flexible manner for the activation of nearby motoneuron pools.     

Environmental changes affect the assembly of soil bacterial community primarily by mediating stochastic processes

Both ‘species fitness difference’‐based deterministic processes, such as competitive exclusion and environmental filtering, and ‘species fitness difference’‐independent stochastic processes, such as birth/death and dispersal/colonization, can influence the assembly of soil microbial communities. However, how both types of processes are mediated by anthropogenic environmental changes has rarely been explored. Here we report a novel and general pattern that almost all anthropogenic environmental changes that took place in a grassland ecosystem affected soil bacterial community assembly primarily through promoting or restraining stochastic processes. We performed four experiments mimicking 16 types of environmental changes and separated the compositional variation of soil bacterial communities caused by each environmental change into deterministic and stochastic components, with a recently developed method. Briefly, because the difference between control and treatment communities is primarily caused by deterministic processes, the deterministic change was quantified as (mean compositional variation between treatment and control) – (mean compositional variation within control). The difference among replicate treatment communities is primarily caused by stochastic processes, so the stochastic change was estimated as (mean compositional variation within treatment) – (mean compositional variation within control). The absolute of the stochastic change was greater than that of the deterministic change across almost all environmental changes, which was robust for both taxonomic and functional‐based criterion. Although the deterministic change may become more important as environmental changes last longer, our findings showed that changes usually occurred through mediating stochastic processes over 5 years, challenging the traditional determinism‐dominated view.     

Nonlinear Time Series Analysis of Nodulation Factor Induced Calcium Oscillations: Evidence for Deterministic Chaos?

Legume plants form beneficial symbiotic interactions with nitrogen fixing bacteria (called rhizobia), with the rhizobia being accommodated in unique structures on the roots of the host plant. The legume/rhizobial symbiosis is responsible for a significant proportion of the global biologically available nitrogen. The initiation of this symbiosis is governed by a characteristic calcium oscillation within the plant root hair cells and this signal is activated by the rhizobia. Recent analyses on calcium time series data have suggested that stochastic effects have a large role to play in defining the nature of the oscillations. The use of multiple nonlinear time series techniques, however, suggests an alternative interpretation, namely deterministic chaos. We provide an extensive, nonlinear time series analysis on the nature of this calcium oscillation response. We build up evidence through a series of techniques that test for determinism, quantify linear and nonlinear components, and measure the local divergence of the system. Chaos is common in nature and it seems plausible that properties of chaotic dynamics might be exploited by biological systems to control processes within the cell. Systems possessing chaotic control mechanisms are more robust in the sense that the enhanced flexibility allows more rapid response to environmental changes with less energetic costs. The desired behaviour could be most efficiently targeted in this manner, supporting some intriguing speculations about nonlinear mechanisms in biological signaling.     

Predators temper the relative importance of stochastic processes in the assembly of prey metacommunities

Communities assemble through a combination of stochastic processes, which can make environmentally similar communities divergent (high β-diversity), and deterministic processes, which can make environmentally similar communities convergent (low β-diversity). Top predators can influence both stochasticity (e.g. colonization and extinction events) and determinism (e.g. size of the realized species pool), in community assembly, and thus their net effect is unknown. We investigated how predatory fish influenced the scaling of prey diversity in ponds at local and regional spatial scales. While fish reduced both local and regional richness, their effects were markedly more intense at the regional scale. Underlying this result was that the presence of fish made localities within metacommunities more similar in their community composition (lower β-diversity), suggesting that fish enhance the deterministic, relative to the stochastic, components of community assembly. Thus, the presence of predators can alter fundamental mechanisms of community assembly and the scaling of diversity within metacommunities.     

Multivariate recurrence plots for visualizing and quantifying the dynamics of spatially extended ecosystems

Few methods for quantifying the dynamics of temporal processes are readily applicable to spatially extended systems when equations governing the motion are unknown. The objective of this paper is to illustrate how the MRP-RQA (multivariate recurrence plot-recurrence quantification analysis) approach may serve to characterize ecosystems driven by both deterministic and stochastic forces. The strength of the MRP-RQA approach resides in its independence from constraining assumptions regarding outliers, noise, stationarity and transients. Its utility is demonstrated by means of two spatiotemporal series (summer and spring datasets) of light intensity variations in an old growth forest ecosystem. Results revealed qualitative differences in homogeneity, transiency, and drift typologies between the MRPs derived from each dataset. RQA estimates of determinism and Kolmogorov entropy supported the idea that mixed chaotic–stochastic dynamics may be common in mesoscale forest habitats. Advantages and inconveniences of the MRP-RQA approach are also discussed in the more general context of monitoring ecosystems.     

Detection of mutual determinism between a pair of spike trains

 A method for detecting mutual deterministic dependence between a pair of spike trains is proposed. When it is assumed that a cell assembly, which is a subgroup of neurons processing a common task, is constituted as a dynamical system, then the mutual determinism between constituent neurons may be directly reflected in functional connectivity in the assembly. The deterministic dependence between two spike trains can be measured with statistical significance using a method of nonlinear prediction. Some examples of simulations are demonstrated in both deterministic and stochastic cases. Received: 8 August 1999 / Accepted in revised form: 29 March 2001