Fractal dimensionality is accepted as a measure of complexity for systems that cannot be described by integer dimensions. However, fractal control mechanisms, physical implications, and relations to nonlinear dynamics have not yet been fully clarified. Herein we explore these issues in a spacetime using a nonlinear integrated model derived by applying Newton’s second law into self-regulating systems. We discover that (i) a stochastic stable fixed point exhibits self-similarity and long-term memory, while a deterministic stable fixed point usually only exhibits self-similarity, if our observation scale is large enough; (ii) stochastic/deterministic period cycles and chaos only exhibit long-term memory, but also self-similarity for even restorative delays; (iii) fractal level of a stable fixed point is controlled primarily by the wave indicators that reflect the relative strength of extrinsic to intrinsic forces: a larger absolute slope (smaller amplitude) indicator leads to higher positive dependence (self-similarity), and a relatively large amplitude indicator or an even restorative delay could make the dependence oscillate; and (iv) fractal levels of period cycles and chaos rely on the intrinsic resistance, restoration, and regulative delays. Our findings suggest that fractals of self-regulating systems can be measured by integer dimensions.

the control mechanisms of heart rate dynamics in a new heart rate nonlinear time series model Zonglu He the control mechanisms and implications of heart rate variability (HRV) under the sympathetic (SnS) and parasympathetic nervous system (pnS) modulation remain poorly understood. Here, we establish the HR model/HRV responder using a nonlinear process derived from newton's second law in stochastic self-restoring systems through dynamic analysis of physiological properties. We conduct model validation by testing, predictions, simulations, and sensitivity and time-scale analysis. We confirm that the outputs of the HRV responder can be accepted as the real data-generating process. Empirical studies show that the dynamic control mechanism of heart rate is a stable fixed point, rather than a strange attractor or transitions between a fixed point and a limit cycle; HR slope (amplitude) may depend on the ratio of cardiac disturbance or metabolic demand mean (standard deviation) to myocardial electrical resistance (PNS-SNS activity). For example, when metabolic demands remain unchanged, HR amplitude depends on PNS to SNS activity; when autonomic activity remains unchanged, HR amplitude during resting reflects basal metabolism. HR parameter alterations suggest that age-related decreased HRV, ultrareduced HRV in heart failure, and ultraelevated HRV in St segment alterations refer to age-related decreased basal metabolism, impaired myocardial metabolism, and SnS hyperactivity triggered by myocardial ischemia, respectively.Cardiac disturbances and PNS-SNS restoring force. Heart rate is determined intrinsically by the rate of spontaneous depolarization at the sinoatrial node, but is also modulated by both sympathetic and parasympathetic efferent innervation in response to cardiac disturbances (physical demands, stress, or hormonal factors) 39 . A cardiac disturbance can be driven by an excitatory event, an inhibitory event, or white noise. Excitatory events include acute stress such as low oxygen, high carbon dioxide, ischemia, or hypotension. Inhibitory events include acute stress such as hypertension or certain physiological states such as rest, sleep, comatose, or anesthetic state. Peripheral chemoreceptors located in the aorta, carotid arteries, and the brain are sensory extensions of the peripheral nervous system into blood vessels by which they detect changes in the concentrations of blood borne chemicals and afferent nerves carry them to the brainstem 40 . When baroreceptors located in the carotid sinus and in the aortic arch are excited by a stretch of the blood vessel, they sense the blood pressure changes and relay them to the lower brainstem. The SNS connected to the heart speeds up a slower-than-normal heartbeat by releasing neurohormones known as catecholamines (epinephrine and norepinephrine). The PNS located in the brainstem and upper or sacral portion of the spinal cord slows down a faster-than-normal heartbeat by releasing the neurohormone acetylcholine. The SNS and PNS exerts excitatory and inhibitory effect...

The mechanisms underlying an effective propagation of high intensity information over a background of irregular firing and response latency in cognitive processes remain unclear. Here we propose a SSCCPI circuit to address this issue. We hypothesize that when a high-intensity thalamic input triggers synchronous spike events (SSEs), dense spikes are scattered to many receiving neurons within a cortical column in layer IV, many sparse spike trains are propagated in parallel along minicolumns at a substantially high speed and finally integrated into an output spike train toward or in layer Va. We derive the sufficient conditions for an effective (fast, reliable, and precise) SSCCPI circuit: (i) SSEs are asynchronous (near synchronous); (ii) cortical columns prevent both repeatedly triggering SSEs and incorrectly synaptic connections between adjacent columns; and (iii) the propagator in interneurons is temporally complete fidelity and reliable. We encode the membrane potential responses to stimuli using the non-linear autoregressive integrated process derived by applying Newton's second law to stochastic resilience systems. We introduce a multithreshold decoder to correct encoding errors. Evidence supporting an effective SSCCPI circuit includes that for the condition, (i) time delay enhances SSEs, suggesting that response latency induces SSEs in high-intensity stimuli; irregular firing causes asynchronous SSEs; asynchronous SSEs relate to healthy neurons; and rigorous SSEs relate to brain disorders. For the condition (ii) neurons within a given minicolumn are stereotypically interconnected in the vertical dimension, which prevents repeated triggering SSEs and ensures signal parallel propagation; columnar segregation avoids incorrect synaptic connections between adjacent columns; and signal propagation across layers overwhelmingly prefers columnar direction. For the condition (iii), accumulating experimental evidence supports temporal transfer precision with millisecond fidelity and reliability in interneurons; homeostasis supports a stable fixed-point encoder by regulating changes to synaptic size, synaptic strength, and ion channel function in the membrane; together all-or-none modulation, active backpropagation, additive effects of graded potentials, and response variability functionally support the multithreshold decoder; our simulations demonstrate that the encoder-decoder is temporally complete fidelity and reliable in special intervals contained within the stable fixed-point range. Hence, the SSCCPI circuit provides a possible mechanism of effective signal propagation in cortical networks.

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