Efficient Maximum Likelihood Estimation of Kinetic Rate Constants from Macroscopic Currents

A new method is described that accurately estimates kinetic constants, conductance and number of ion channels from macroscopic currents. The method uses both the time course and the strength of correlations between different time points of macroscopic currents and utilizes the property of semiseparability of covariance matrix for computationally efficient estimation of current likelihood and its gradient. The number of calculation steps scales linearly with the number of channel states as opposed to the cubic dependence in a previously described method. Together with the likelihood gradient evaluation, which is almost independent of the number of model parameters, the new approach allows evaluation of kinetic models with very complex topologies. We demonstrate applicability of the method to analysis of synaptic currents by estimating accurately rate constants of a 7-state model used to simulate GABAergic macroscopic currents

Inositol Trisphosphate and Cyclic ADP-Ribose–Mediated Release of Ca2+ from Single Isolated Pancreatic Zymogen Granules

AbstractIn pancreatic acinar cells low (physiological) agonist concentrations evoke cytosolic Ca2+ spikes specifically in the apical secretory pole that contains a high density of secretory (zymogen) granules (ZGs). Inositol 1,4,5-trisphosphate (IP3) is believed to release Ca2+ from the endoplasmic reticulum, but we have now tested whether the Ca2+-releasing messengers IP3 and cyclic ADP-ribose (cADPr) can liberate Ca2+ from ZGs. In experiments on single isolated ZGs, we show using confocal microscopy that IP3 and cADPr evoke a marked decrease in the free intragranular Ca2+ concentration. Using a novel high resolution method, we have measured changes in the Ca2+ concentration in the vicinity of an isolated ZG and show that IP3 and cADPr cause rapid Ca2+ release from the granule, explaining the agonist-evoked cytosolic Ca2+ rise in the secretory pole

Segmental and descending control of primary afferent input to the spinal lamina X

Despite being involved in a number of functions, such as nociception and locomotion, spinal lamina X remains one of the least studied central nervous system regions. Here, we show that Aδ- and C-afferent inputs to lamina X neurons are presynaptically inhibited by homo- and heterosegmental afferents as well as by descending fibers from the corticospinal tract, dorsolateral funiculus, and anterior funiculus. Activation of descending tracts suppresses primary afferent-evoked action potentials and also elicits excitatory (mono- and polysynaptic) and inhibitory postsynaptic responses in lamina X neurons. Thus, primary afferent input to lamina X is subject to both spinal and supraspinal control being regulated by at least 5 distinct pathways

A Model for the Fast Synchronous Oscillations of Firing Rate in Rat Suprachiasmatic Nucleus Neurons Cultured in a Multielectrode Array Dish

<div><p>When dispersed and cultured in a multielectrode dish (MED), suprachiasmatic nucleus (SCN) neurons express fast oscillations of firing rate (FOFR; fast relative to the circadian cycle), with burst duration ∼10 min, and interburst interval varying from 20 to 60 min in different cells but remaining nevertheless rather regular in individual cells. In many cases, separate neurons in distant parts of the 1 mm recording area of a MED exhibited correlated FOFR. Neither the mechanism of FOFR nor the mechanism of their synchronization among neurons is known. Based on recent data implicating vasoactive intestinal polypeptide (VIP) as a key intercellular synchronizing agent, we built a model in which VIP acts as both a feedback regulator to generate FOFR in individual neurons, and a diffusible synchronizing agent to produce coherent electrical output of a neuronal network. In our model, VIP binding to its (VPAC₂) receptors acts through G_s G-proteins to activate adenylyl cyclase (AC), increase intracellular cAMP, and open cyclic-nucleotide-gated (CNG) cation channels, thus depolarizing the cell and generating neuronal firing to release VIP. In parallel, slowly developing homologous desensitization and internalization of VPAC₂ receptors terminates elevation of cAMP and thereby provides an interpulse silent interval. Through mathematical modeling, we show that this VIP/VPAC₂/AC/cAMP/CNG-channel mechanism is sufficient for generating reliable FOFR in single neurons. When our model for FOFR is combined with a published model of synchronization of circadian rhythms based on VIP/VPAC₂ and Per gene regulation synchronization of circadian rhythms is significantly accelerated. These results suggest that (a) auto/paracrine regulation by VIP/VPAC₂ and intracellular AC/cAMP/CNG-channels are sufficient to provide robust FOFR and synchrony among neurons in a heterogeneous network, and (b) this system may also participate in synchronization of circadian rhythms.</p></div

Induction of the oscillations in the heterogeneous population of SCN neurons by a small group of oscillating cells.

<p>Network of 20 cells was modeled, with 4 cells having default parameter set and other 16 cells having CNG conductance twice lower than default and all other parameters as default. <b>A</b>. Firing rate in the population before introduction of exchange by VIP. Only 4 abovementioned cells generate oscillations of firing rate. <b>B</b>. Firing rate in the population after the exchange by VIP was introduced (D = 0.5). All 20 cells started to oscillate in synchrony. This result was robust with respect to CNG channel conductance and moderate random perturbations of parameters. 4 cells with default parameters were capable to induce synchronous oscillations in the remaining 16 cells with reduced (20% of default value) CNG channel conductance. All other parameters of the model were selected for each of these 16 cells randomly and uniformly from the interval [p0-0.05*p0; p0+0.05*p0], where p0 is default value of each parameter. <b>C, E</b>. Firing rate and phase distribution in the population of SCN neurons after introduction of weak coupling through VIP exchange (D = 0.3). <b>D, F</b>. The same plots for strongly coupled oscillators (D = 6).</p

Basic properties of VIP-cAMP signaling in the model of FOFR.

<p><b>A</b>. Response of the model neuron to the application of different VIP concentration steps (from 0.1 nM to 1 µM). Shown are concentration of cAMP (nM) and the ratio of the number of membrane VPAC₂ receptors to the total number of VPAC₂ receptors. For comparison, experimental data from Murthy et al. (Figure 10 in <a href="http://www.plosone.org/article/info:doi/10.1371/journal.pone.0106152#pone.0106152-Murthy1" target="_blank">[25]</a>) describing ¹²⁵I-VIP binding to surface VPAC₂ receptors during and after VIP (1 µM) application are shown by open circles. <b>B</b>. Dependencies of adenylate cyclase (AC) activity (measured as % of maximal concentration of G_αs-AC complexes) on VIP level before (smooth line) and after (dotted line) desensitization and internalization of VPAC₂ receptors induced by the application of VIP (1 µM) for 30 min.</p

Mechanism of oscillatory activity in the model.

<p><b>A</b>. Firing rate and concentration of the key molecular players during FOFR. <b>B</b>. Dynamics of the model in the VIP – VPAC₂ plane in the case of a very slow rate of recovery of membrane VPAC₂ receptor concentration after internalization of these receptors (k₇₁ was set to 10⁻⁸ s⁻¹, see <a href="http://www.plosone.org/article/info:doi/10.1371/journal.pone.0106152#pone.0106152.s001" target="_blank">Text S1</a>). VPAC₂ nullcline is shown in green, VIP nullcline is shown in red. A stable limit cycle is shown in blue. The range of the nullclines intersection in which stable oscillations were observed is shown in yellow. <b>C</b>. Dynamics of the model in the VIP – cAMP plane under fixed VPAC₂ membrane concentration. VIP nullcline – green, cAMP nullcline – red, stable equilibrium – closed circles, unstable equilibrium – open circle. <b>D</b>. Dynamics of the model in the VIP – VPAC₂ plane with experimentally observed rate of recovery from receptors internalization. All notations are the same as in B.</p

Parameter sensitivity of the FOFR model.

<p><b>A</b>. The ranges of parameters for which oscillations were observed. Initial concentrations of all molecules in the model, a key kinetic rates and some parameters from those that we have added to the model of Hao et al. <a href="http://www.plosone.org/article/info:doi/10.1371/journal.pone.0106152#pone.0106152-Hao1" target="_blank">[19]</a>, were sequentially varied in the range [x0/50, x0*50] (where x0 is default initial concentration). 15 hours of the model state evolution was simulated, and the parameter ranges were found in which the oscillations of cAMP concentration with the amplitude of more than 10% of the cAMP average level was observed. Only those parameter ranges (blue bars) are shown for which oscillations disappeared during the parameter variation in a range smaller than [x0/50, x0*50], where x0 is default initial value of each parameter (shown in green, scale is logarithmic). <b>B-G</b>. Bifurcation diagrams of the equilibrium (magenta line) for a key model parameters. In addition to the equilibrium, the range of stable limit cycle is shown (green lines). Dependencies of the period of oscillations on each of the model parameters are shown in the inset.</p

Colocalization of VPAC₂ receptors and CNG channels improve synchronization of the circadian activity in the network of SCN neurons.

<p>A₁. Firing rates (top) and Per gene expression (bottom) before and after introduction of VIP exchange (red vertical line) in the network of 10 cells. The firing rate was regulated directly by expression of circadian genes (Model without VIP/CNG coupling, 1^st experiment, see Methods). All cells had default sets of parameters in both circadian clock and FOFR models. Circadian discharges started with phase shifts uniformly distributed within 6 h interval and without exchange by VIP (D = 0) up to the 91 hour (red vertical line). Then VIP exchange between cells was introduced (D = 0.5). A₂. The same variables after stabilization of circadian oscillations of firing rate and Per gene expression in heterogeneous population of coupled oscillators before (D = 0.5) and after (D = 0) their uncoupling (Model without VIP/CNG, 2^nd experiment). All cells had default sets of parameters of fast oscillations but circadian clock parameters were randomly and uniformly perturbed within 3.5% interval around their default values. A₃. Scheme of the respective Model without VIP/CNG coupling. B₁, B₂. The results of the same experiments as in A₁, A₂ for the Model with VIP/CNG coupling with FOFR, i.e. when firing rate was mostly regulated by external VIP via CNG channels colocalized with VPAC₂ receptors, but with influence of Per gene product on the conductance of the minor fraction of CNG channels (see Methods). B₃ Scheme of the respective Model with VIP/CNG coupling with FOFR. C₁, C₂. The same experiments as in A, B, for the Model with VIP/CNG coupling without FOFR (i.e. when firing rate was influenced by both fast and slow VIP signaling loops similar to the Model with VIP/CNG coupling with FOFR, but there were no FOFR, see Methods) C₃. Scheme of the respective Model with VIP/CNG coupling without FOFR.</p

Synchronization of the oscillations in heterogeneous population of SCN neurons.

<p>The model included 50 SCN neurons that initially oscillated independently. Initial concentration of each of the molecular species in the model was selected randomly from the interval [x0−0.2*x0; x0+0.2*x0], and each rate constant – from the interval [k0−0.2*k0; k0+0.2*k0], where x0 and k0 are default values for initial concentration and kinetic rate for each molecule and chemical transition in the model, respectively. <b>A</b>. Firing rates for 10 of 50 SCN neurons in the population before and after enabling VIP exchange between the neurons. A vertical red line marks the time when intercellular exchange by VIP was switched on (D = 0.5, Figs. 5A, 7A₁, B₁, C₁) and off (D = 0, Figs. 5A, 7A₂, B₂, C₂). <b>B, C</b>. Distribution of oscillation phases before (B) and after (C) introduction of VIP exchange.</p