Stochasticity in Ca2+ increase in spines enables robust and sensitive information coding

Dendritic spine morphology regulates calcium-dependent synaptic weight change

Dendritic spines act as biochemical computational units and must adapt their responses according to their activation history. Calcium influx acts as the first signaling step during postsynaptic activation and is a determinant of synaptic weight change. Dendritic spines also come in a variety of sizes and shapes. To probe the relationship between calcium dynamics and spine morphology, we used a stochastic reaction-diffusion model of calcium dynamics in idealized and realistic geometries. We show that despite the stochastic nature of the various calcium channels, receptors, and pumps, spine size and shape can modulate calcium dynamics and subsequently synaptic weight updates in a deterministic manner. Through a series of exhaustive simulations and analyses, we found that the calcium dynamics and synaptic weight change depend on the volume-to-surface area of the spine. The relationships between calcium dynamics and spine morphology identified in idealized geometries also hold in realistic geometries, suggesting that there are geometrically determined deterministic relationships that may modulate synaptic weight change.

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

Calcium (Ca²⁺) is a second messenger assumed to control changes in synaptic strength in the form of both long-term depression and long-term potentiation at Purkinje cell dendritic spine synapses via inositol trisphosphate (IP₃)-induced Ca²⁺ release. These Ca²⁺ transients happen in response to stimuli from parallel fibers (PFs) from granule cells and climbing fibers (CFs) from the inferior olivary nucleus. These events occur at low numbers of free Ca²⁺, requiring stochastic single-particle methods when modeling them. We use the stochastic particle simulation program MCell to simulate Ca²⁺ transients within a three-dimensional Purkinje cell dendritic spine. The model spine includes the endoplasmic reticulum, several Ca²⁺ transporters, and endogenous buffer molecules. Our simulations successfully reproduce properties of Ca²⁺ transients in different dynamical situations. We test two different models of the IP₃ receptor (IP₃R). The model with nonlinear concentration response of binding of activating Ca²⁺ reproduces experimental results better than the model with linear response because of the filtering of noise. Our results also suggest that Ca²⁺-dependent inhibition of the IP₃R needs to be slow to reproduce experimental results. Simulations suggest the experimentally observed optimal timing window of CF stimuli arises from the relative timing of CF influx of Ca²⁺ and IP₃ production sensitizing IP₃R for Ca²⁺-induced Ca²⁺ 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 Ca²⁺ transients under ataxia.

Robustness against additional noise in cellular information transmission

Fluctuations in intracellular reactions (intrinsic noise) reduce the information transmitted from an extracellular input to a cellular response. However, recent studies have demonstrated that the decrease in the transmitted information with respect to extracellular input fluctuations (extrinsic noise) is smaller when the intrinsic noise is larger. Therefore, it has been suggested that robustness against extrinsic noise increases with the level of the intrinsic noise. We call this phenomenon intrinsic noise-induced robustness (INIR). As previous studies on this phenomenon have focused on complex biochemical reactions, the relation between INIR and the input-output of a system is unclear. Moreover, the mechanism of INIR remains elusive. In this paper, we address these questions by analyzing simple models. We first analyze a model in which the input-output relation is linear. We show that the robustness against extrinsic noise increases with the intrinsic noise, confirming the INIR phenomenon. Moreover, the robustness against the extrinsic noise is more strongly dependent on the intrinsic noise when the variance of the intrinsic noise is larger than that of the input distribution. Next, we analyze a threshold model in which the output depends on whether the input exceeds the threshold. When the threshold is equal to the mean of the input, INIR is realized, but when the threshold is much larger than the mean, the threshold model exhibits stochastic resonance, and INIR is not always apparent. The robustness against extrinsic noise and the transmitted information can be traded off against one another in the linear model and the threshold model without stochastic resonance, whereas they can be simultaneously increased in the threshold model with stochastic resonance.

NMDAR-Mediated Ca2+ Increase Shows Robust Information Transfer in Dendritic Spines

A dendritic spine is a small structure on the dendrites of a neuron that processes input timing information from other neurons. Tens of thousands of spines are present on a neuron. Why are spines so small and many? We addressed this issue by using the stochastic simulation model of N-methyl-D-aspartate receptor (NMDAR)-mediated Ca²⁺ increase. NMDAR-mediated Ca²⁺ increase codes the input timing information between prespiking and postspiking. We examined how much the input timing information is encoded by Ca²⁺ increase against presynaptic fluctuation. We found that the input timing information encoded in the cell volume (10³μm³) largely decreases against the presynaptic fluctuation, whereas that in the spine volume (10^-1μm³) slightly decreases. Therefore, the input timing information encoded in the spine volume is more robust against presynaptic fluctuation than that in the cell volume. We further examined the mechanism of the robust information transfer in the spine volume. We demonstrated that the condition for the robustness is that the stochastic NMDAR-mediated Ca²⁺ increase (intrinsic noise) becomes much larger than the presynaptic fluctuation (extrinsic noise). When the presynaptic fluctuation is large, the condition is satisfied in the spine volume but not in the cell volume. Moreover, we compared the input timing information encoded in many small spines with that encoded in a single large spine. We found that the input timing information encoded in many small spines are larger than that in a single large spine when presynaptic fluctuation is large because of their robustness. Thus, robustness is a functional reason why dendritic spines are so small and many.

Motif analysis for small-number effects in chemical reaction dynamics

The number of molecules involved in a cell or subcellular structure is sometimes rather small. In this situation, ordinary macroscopic-level fluctuations can be overwhelmed by non-negligible large fluctuations, which results in drastic changes in chemical-reaction dynamics and statistics compared to those observed under a macroscopic system (i.e., with a large number of molecules). In order to understand how salient changes emerge from fluctuations in molecular number, we here quantitatively define small-number effect by focusing on a "mesoscopic" level, in which the concentration distribution is distinguishable both from micro- and macroscopic ones and propose a criterion for determining whether or not such an effect can emerge in a given chemical reaction network. Using the proposed criterion, we systematically derive a list of motifs of chemical reaction networks that can show small-number effects, which includes motifs showing emergence of the power law and the bimodal distribution observable in a mesoscopic regime with respect to molecule number. The list of motifs provided herein is helpful in the search for candidates of biochemical reactions with a small-number effect for possible biological functions, as well as for designing a reaction system whose behavior can change drastically depending on molecule number, rather than concentration.

Small-Volume Effect Enables Robust, Sensitive, and Efficient Information Transfer in the Spine

Why is the spine of a neuron so small that it can contain only small numbers of molecules and reactions inevitably become stochastic? We previously showed that, despite such noisy conditions, the spine exhibits robust, sensitive, and efficient features of information transfer using the probability of Ca²⁺ increase; however, the mechanisms are unknown. In this study, we show that the small volume effect enables robust, sensitive, and efficient information transfer in the spine volume, but not in the cell volume. In the spine volume, the intrinsic noise in reactions becomes larger than the extrinsic noise of input, resulting in robust information transfer despite input fluctuation. In the spine volume, stochasticity makes the Ca²⁺ increase occur with a lower intensity of input, causing higher sensitivity to lower intensity of input. The volume-dependency of information transfer increases its efficiency in the spine volume. Thus, we propose that the small-volume effect is the functional reason why the spine has to be so small.