Stochastic reaction-diffusion modeling of calcium dynamics in 3D-dendritic spines of purkinje cells

Calcium (Ca(2+)) is a second messenger assumed to control changes in synaptic strength in the form of both long-term depression (LTD) and long-term potentiation (LTP) at Purkinje cell dendritic spine synapses via inositol trisphosphate (IP(3)) induced Ca(2+) release. These Ca(2+) transients happen in response to stimuli from parallel fibers (PF) from granule cells and climbing fibers (CF) from the inferior olivary nucleus. These events occur at low numbers of free Ca(2+) requiring stochastic single particle methods when modeling them. We use the stochastic particle simulation program MCell to simulate Ca(2+) transients within a three-dimensional Purkinje cell dendritic spine. The model spine includes the endoplasmic reticulum (ER), several Ca(2+) transporters, and endogenous buffer molecules. Our simulations successfully reproduce properties of Ca(2+) transients in different dynamical situations. We test two different models of the IP(3) receptor (IP3R). The model with non-linear concentration response of binding of activating Ca(2+) reproduces experimental results better than the model with linear response due to the filtering of noise. Our results also suggest that Ca(2+) dependent inhibition of the IP3R needs to be slow in order to reproduce experimental results. Simulations suggest the experimentally observed optimal timing window of CF stimuli to arise from the relative timing of CF influx of Ca(2+) and IP(3) production sensitizing IP3R for Ca(2+) induced Ca(2+) release. We also model Ataxia, a loss of fine motor control assumed to be the result of malfunctioning information transmission at the granule to Purkinje cell synapse, resulting in a decrease or loss of Ca2+ transients. Finally, we propose possible ways of recovering Ca(2+) transients under Ataxia

Modeling IP(3) induced Ca(2+) signaling based on its interspike interval statistics

Inositol 1,4,5-trisphosphate (IP(3)) induced Ca(2+) signaling is a second messenger system used by almost all eukaryotic cells. Recent research identified 8 general properties of Ca(2+) spiking common to all cell types investigated and demonstrated randomness of Ca(2+) signaling on all structural levels. We suggest a theory of Ca(2+) spiking starting from the random behaviour of IP(3) receptor channel clusters mediating the release of Ca(2+) from the endoplasmic reticulum. Spike generation begins after the absolute refractory period of the previous spike. According to its hierarchical spreading from initiating channel openings to cell level, we describe it as a first passage process from none to all clusters open while the cell recovers from the inhibition which terminated the previous spike. Our theory reproduces quantitatively all general properties for different IP(3) pathways including the exponential stimulation response relation of the average interspike interval (ISI) T(av) and its robustness properties, random spike timing with a linear moment relation between T(av) and the ISI standard deviation and its robustness properties, sensitive dependency of T(av) on diffusion properties, and non-oscillatory local dynamics. We explain large cell variability of T(av) observed in experiments by variability of channel cluster coupling by Ca(2+) induced Ca(2+) release, the number of clusters and IP(3) pathway components expression levels. We predict the relation between puff probability and agonist concentration, and [IP(3)] and agonist concentration. Differences of spike behaviour between cell types and stimulating agonists are explained by the different types of negative feedback terminating spikes. In summary, the hierarchical random character of spike generation explains all of the identified general properties

The stretch to stray on time: resonant length of random walks in a transient

First-passage times in random walks have a vast number of diverse applications in physics, chemistry, biology, and finance. In general, environmental conditions for a stochastic process are not constant on the time scale of the average first-passage time, or control might be applied to reduce noise. We investigate moments of the first-passage time distribution under a transient describing relaxation of environmental conditions. We solve the Laplace-transformed (generalized) master equation analytically using a novel method that is applicable to general state schemes. The first-passage time from one end to the other of a linear chain of states is our application for the solutions. The dependence of its average on the relaxation rate obeys a power law for slow transients. The exponent ν depends on the chain length N like ν=−N/(N+1) to leading order. Slow transients substantially reduce the noise of first-passage times expressed as the coefficient of variation (CV), even if the average first-passage time is much longer than the transient. The CV has a pronounced minimum for some lengths, which we call resonant lengths. These results also suggest a simple and efficient noise control strategy, and are closely related to the timing of repetitive excitations, coherence resonance and information transmission by noisy excitable systems. A resonant number of steps from the inhibited state to the excitation threshold and slow recovery from negative feedback provide optimal timing noise reduction and information transmission

Modeling IP(3)-induced Ca(2+) signaling based on its interspike interval statistics

Inositol 1,4,5-trisphosphate (IP(3)) induced Ca(2+) signaling is a second messenger system used by almost all eukaryotic cells. Recent research demonstrated randomness of Ca(2+) signaling on all structural levels. We compile 8 general properties of Ca(2+) spiking common to all cell types investigated and suggest a theory of Ca(2+) spiking starting from the random behavior of IP(3) receptor channel clusters mediating the release of Ca(2+) from the endoplasmic reticulum capturing all general properties and pathway specific behavior. Spike generation begins after the absolute refractory period of the previous spike. According to its hierarchical spreading from initiating channel openings to cell level, we describe it as a first passage process from none to all clusters open while the cell recovers from the inhibition which terminated the previous spike. Our theory reproduces the exponential stimulation response relation of the average interspike interval (ISI) T(av) and its robustness properties, random spike timing with a linear moment relation between T(av) and the ISI standard deviation and its robustness properties, sensitive dependency of T(av) on diffusion properties, and non-oscillatory local dynamics. We explain large cell variability of T(av) observed in experiments by variability of channel cluster coupling by Ca(2+) induced Ca(2+) release, the number of clusters and IP(3) pathway components expression levels. We predict the relation between puff probability and agonist concentration, and [IP(3)] and agonist concentration. Differences of spike behavior between cell types and stimulating agonists are explained by the different types of negative feedback terminating spikes. In summary, the hierarchical random character of spike generation explains all of the identified general properties