Volume-dependent time window of Ca²⁺ increases.

<p><b>A–C</b>, Distributions of the Ca²⁺ response along the PF-CF interval (left) in 0.1 µm³ (a spine volume) (<b>A</b>), in 5,000 µm³ (a cell volume) (<b>B</b>), and in 10 µm³ (<b>C</b>). The probability components (middle) and the amplitude components (right) are also shown. The colors in the left and right panels code the probability of the Ca²⁺ response at the indicated PF-CF interval. Arrowheads in the left panels indicate the thresholds between two peaks of the bimodal marginal distributions. The probability components denote the frequencies of the Ca²⁺ response above/below the thresholds (solid/dashed lines, respectively), and the amplitude components denote the Ca²⁺ response above/below the thresholds (indicated by solid/dashed braces, respectively). The PF-CF interval was defined by the time difference between the first PF and CF inputs. <b>D</b>, Volume-dependency of the input timing information coded by the total distribution of the Ca²⁺ response (black), by the probability component (red), and by the amplitude component (blue). <b>E</b>, Volume-dependency of the input timing information per volume. <b>F</b>, Relative contribution of the probability (red) and amplitude (blue) component to the input timing information.</p

Stochasticity in Ca²⁺ Increase in Spines Enables Robust and Sensitive Information Coding

<div><p>A dendritic spine is a very small structure (∼0.1 µm³) of a neuron that processes input timing information. Why are spines so small? Here, we provide functional reasons; the size of spines is optimal for information coding. Spines code input timing information by the probability of Ca²⁺ increases, which makes robust and sensitive information coding possible. We created a stochastic simulation model of input timing-dependent Ca²⁺ increases in a cerebellar Purkinje cell's spine. Spines used probability coding of Ca²⁺ increases rather than amplitude coding for input timing detection via stochastic facilitation by utilizing the small number of molecules in a spine volume, where information per volume appeared optimal. Probability coding of Ca²⁺ increases in a spine volume was more robust against input fluctuation and more sensitive to input numbers than amplitude coding of Ca²⁺ increases in a cell volume. Thus, stochasticity is a strategy by which neurons robustly and sensitively code information.</p></div

Robust and sensitive probability coding in the stochastic model.

<p><b>A</b>, Distribution of the Ca²⁺ response in a single spine (upper panels) and in a cell (lower panels) due to stimulation of PF and CF inputs with fluctuating the PF amplitudes. CVs of the PF amplitudes were 0.1 (left panels) and 0.5 (right panels). <b>B</b>, Input timing information per volume, coded by the Ca²⁺ response, in a spine (red) and in a cell (black). <b>C</b>, Distribution of the Ca²⁺ response in a single spine (upper panels) and in a cell (lower panels) with a single PF input (left panels), three PF inputs (middle panels), seven PF inputs (right panels). In a spine, large Ca²⁺ increase were observed in a few trials in response to three PF inputs (white arrowhead). CV of the PF amplitudes was 0.1. <b>D</b>, Input timing information per volume in a spine (red) and in a cell (black).</p

Ca²⁺ increase in cerebellar Purkinje cells.

<p><b>A</b>, Cerebellar Purkinje cells. Male mouse cerebellar Purkinje cells were doubly stained with anti-calbindin antibody to visualize whole cells (green) and with anti-mGluR (metabotropic glutamate receptor) antibody to specifically visualize the spines of the PF-Purkinje cell synapses (red). The inset in the left image is magnified in the right panel. The average volume of spines in cerebellar Purkinje cells has been reported to be 0.1 µm³, which is 10⁴-fold smaller than a cell body (5,000 µm³). White circles in the left and right panels indicate a typical soma and spine, respectively. <b>B</b>, Schematic representation of PF and CF inputs-dependent Ca²⁺ increase in the simulation model <a href="http://www.plosone.org/article/info:doi/10.1371/journal.pone.0099040#pone.0099040-Doi1" target="_blank">[14]</a> (<a href="http://www.plosone.org/article/info:doi/10.1371/journal.pone.0099040#s4" target="_blank">Materials and Methods</a>). Abbreviations: Glu; glutamate, mGluR; metabotropic glutamate receptor, IP₃; inositol trisphosphate; AMPAR; α-amino-3-hydroxy-5-methyl-4-isoxazole propionic acid receptor, VGCC; voltage-gated Ca²⁺ channels. Parentheses indicate initial numbers of the indicated molecules in the stochastic model. <b>C</b>, Ca²⁺ responses along the PF-CF interval in the experiments (black circles) <a href="http://www.plosone.org/article/info:doi/10.1371/journal.pone.0099040#pone.0099040-Wang1" target="_blank">[12]</a>, and mean Ca²⁺ responses in the stochastic simulation in a spine volume (0.1 µm³) (solid line) and in the stochastic simulation in a cell volume (5,000 µm³) (dashed line). The Ca²⁺ response was defined as the average relative change in fluorescence (ΔF/F₀) of Magnesium Green 1, a Ca²⁺ indicator <a href="http://www.plosone.org/article/info:doi/10.1371/journal.pone.0099040#pone.0099040-Doi1" target="_blank">[14]</a>. Positive sign of the interval are given to Δ<i>t</i> msec when PF inputs precede CF inputs; otherwise, a negative sign is given. Five PF inputs at 100 Hz and a single CF were given.</p

Ca²⁺ increases in the stochastic model.

<p><b>A, B</b>, Ca²⁺ increase due to stimulation of PF and CF inputs with Δt = 160 msec (<b>A</b>) and Δt = −400 msec (<b>B</b>) in the stochastic model in a spine volume (gray lines, n = 2,000 for each timing, 20 examples are shown) and in a cell volume (black lines, n = 20 for each timing). <b>C</b>, Distributions of the Ca²⁺ response in a spine volume with Δt = 160 msec (solid line) and Δt = −400 msec (dashed line). Black and white arrowheads indicate the means of the Ca²⁺ response in a cell volume with Δt = 160 msec and Δt = −400 msec, respectively. Note that Ca²⁺ increases in a cell volume were almost the same between trials, leading to the overlap of the time courses (<b>A</b>, <b>B</b>). <b>D</b>, The distribution of the Ca²⁺ response (upper panel) was divided into the distribution with Ca²⁺ spikes (<i>s</i> = 1, black) and the distribution without Ca²⁺ spikes (<i>s</i> = 0, gray with dashed line). Then, each right (<i>s</i> = 1) and left (<i>s</i> = 0) distribution is decomposed into the probability components, which is probability of Ca²⁺ spiking (middle panel), and amplitude components, which is distribution of the Ca²⁺ response conditioned by Ca²⁺ spiking (lower panel). See <a href="http://www.plosone.org/article/info:doi/10.1371/journal.pone.0099040#s4" target="_blank">materials and methods</a> for the detailed descriptions.</p

Decrease of the input timing information with the change of the CV of PF amplitude from 0.05 to 0.5 and the number of PF inputs from 5 to 3.

<p>Decrease of the input timing information with the change of the CV of PF amplitude from 0.05 to 0.5 and the number of PF inputs from 5 to 3.</p

The nonlinear ARX model of the IEGs.

<p>(A) The simulation result of the nonlinear ARX model (solid lines) together with the experimental results in <a href="http://www.plosone.org/article/info:doi/10.1371/journal.pone.0057037#pone-0057037-g001" target="_blank">Figure 1B</a> (dots). The colour codes are the same as in <a href="http://www.plosone.org/article/info:doi/10.1371/journal.pone.0057037#pone-0057037-g001" target="_blank">Figure 1B</a>. The experimental data in <a href="http://www.plosone.org/article/info:doi/10.1371/journal.pone.0057037#pone-0057037-g001" target="_blank">Figure 1B</a> and <a href="http://www.plosone.org/article/info:doi/10.1371/journal.pone.0057037#pone.0057037.s002" target="_blank">Figure S2</a> were used for parameter estimation of the nonlinear ARX model. (B) The identified systems by the nonlinear ARX model. The upstream dependency (selected inputs), Hill functions, and frequency response curve of the nonlinear ARX model were shown. The selected inputs, pERK (solid line), pCREB (dotted line), pJNK (dashed line), and c-FOS (dashed and dotted line) were numbered.</p

Temporal Decoding of MAP Kinase and CREB Phosphorylation by Selective Immediate Early Gene Expression

<div><p>A wide range of growth factors encode information into specific temporal patterns of MAP kinase (MAPK) and CREB phosphorylation, which are further decoded by expression of immediate early gene products (IEGs) to exert biological functions. However, the IEG decoding system remain unknown. We built a data-driven based on time courses of MAPK and CREB phosphorylation and IEG expression in response to various growth factors to identify how signal is processed. We found that IEG expression uses common decoding systems regardless of growth factors and expression of each IEG differs in upstream dependency, switch-like response, and linear temporal filters. Pulsatile ERK phosphorylation was selectively decoded by expression of EGR1 rather than c-FOS. Conjunctive NGF and PACAP stimulation was selectively decoded by synergistic JUNB expression through switch-like response to c-FOS. Thus, specific temporal patterns and combinations of MAPKs and CREB phosphorylation can be decoded by selective IEG expression via distinct temporal filters and switch-like responses. The data-driven modeling is versatile for analysis of signal processing and does not require detailed prior knowledge of pathways.</p> </div

The selective expression of EGR1 in response to pulsatile ERK phosphorylation.

<p>(A) The step (5 ng/ml, red), pulse (5 ng/ml, 6 min, blue), and pulsatile NGF stimulation (0.5 ng/ml, 6 min with 12-min intervals for four times, green) were given as indicated by bars (top), and pERK, pCREB, EGR1, and c-FOS were measured in experiments (dots). Using the experimental data of pERK and pCREB as the selected inputs, the outputs (c-FOS and EGR1) were simulated by the nonlinear ARX model (solid lines). (B) Interval dependency of EGR1 and c-FOS expression. The pulsatile NGF stimulation (0.5 ng/ml, 15-min duration for each pulse) with the indicated intervals were given, and pERK, EGR1, and c-FOS expression were measured in experiments. The area under the curve (AUC) (0–480 min) of EGR1 and c-FOS are shown in bars. The intervals are indicated by the colour codes. Bars represent means ±S.D.(n = 4). Note that 15-min duration of pulses was used in <a href="http://www.plosone.org/article/info:doi/10.1371/journal.pone.0057037#pone-0057037-g004" target="_blank">Figure 4B</a> because of the technical limitation of probe numbers of the automated liquid-handling robots, and pulsatile stimulation with 6-min pulse duration and 12-min intervals were available at most four times (<a href="http://www.plosone.org/article/info:doi/10.1371/journal.pone.0057037#pone-0057037-g004" target="_blank">Figure 4A</a>).</p

System identification by the nonlinear ARX model.

<p>(A) The modeling scheme of the nonlinear ARX model. Upstream dependency was determined by lag order number, <i>m</i>. For example, if <i>m</i> = 0, upstream signal is not transmitted downstream, otherwise signal is transmitted downstream. The signals of the selected upstream molecules were transformed successively by Hill function and linear ARX model, that characterise a system with switch-like (solid line) or graded (dotted line) dose response, and with temporal filters such as a low-pass filter (dotted line) and that with an inverse notch (solid line), respectively (see <a href="http://www.plosone.org/article/info:doi/10.1371/journal.pone.0057037#s2" target="_blank">Materials and methods</a>). (B) Temporal signal transformation in the nonlinear ARX model. For example, signal transformation in the nonlinear ARX model of c-FOS was shown. pERK and pCREB were selected upstream molecules, but pp38 and pJNK were not (<i>m</i> = 0). The signals of pERK and pCREB were transformed by the Hill equations. Then, the transformed signals by the Hill equations were temporally transformed by the linear ARX model. The sum of the transformed signals by the linear ARX model was c-FOS, the final output of the nonlinear ARX model of c-FOS.</p