Spontaneous single-nucleotide substitutions and microsatellite mutations have distinct distributions of fitness effects.

The fitness effects of new mutations determine key properties of evolutionary processes. Beneficial mutations drive evolution, yet selection is also shaped by the frequency of small-effect deleterious mutations, whose combined effect can burden otherwise adaptive lineages and alter evolutionary trajectories and outcomes in clonally evolving organisms such as viruses, microbes, and tumors. The small effect sizes of these important mutations have made accurate measurements of their rates difficult. In microbes, assessing the effect of mutations on growth can be especially instructive, as this complex phenotype is closely linked to fitness in clonally evolving organisms. Here, we perform high-throughput time-lapse microscopy on cells from mutation-accumulation strains to precisely infer the distribution of mutational effects on growth rate in the budding yeast, Saccharomyces cerevisiae. We show that mutational effects on growth rate are overwhelmingly negative, highly skewed towards very small effect sizes, and frequent enough to suggest that deleterious hitchhikers may impose a significant burden on evolving lineages. By using lines that accumulated mutations in either wild-type or slippage repair-defective backgrounds, we further disentangle the effects of 2 common types of mutations, single-nucleotide substitutions and simple sequence repeat indels, and show that they have distinct effects on yeast growth rate. Although the average effect of a simple sequence repeat mutation is very small (approximately 0.3%), many do alter growth rate, implying that this class of frequent mutations has an important evolutionary impact.

Parallel proteomics and phosphoproteomics defines starvation signal specific processes in cell quiescence.

Cells arrest growth and enter a quiescent state upon nutrient deprivation. However, the molecular processes by which cells respond to different starvation signals to regulate exit from the cell division cycle and initiation of quiescence remains poorly understood. To study the role of protein expression and signaling in quiescence we combined temporal profiling of the proteome and phosphoproteome using stable isotope labeling with amino acids in cell culture (SILAC) in Saccharomyces cerevisiae (budding yeast). We find that carbon and phosphorus starvation signals activate quiescence through largely distinct remodeling of the proteome and phosphoproteome. However, increased expression of mitochondrial proteins is associated with quiescence establishment in response to both starvation signals. Deletion of the putative quiescence regulator RIM15, which encodes a serine-threonine kinase, results in reduced survival of cells starved for phosphorus and nitrogen, but not carbon. However, we identified common protein phosphorylation roles for RIM15 in quiescence that are enriched for RNA metabolism and translation. We also find evidence for RIM15-mediated phosphorylation of some targets, including IGO1, prior to starvation consistent with a functional role for RIM15 in proliferative cells. Finally, our results reveal widespread catabolism of amino acids in response to nitrogen starvation, indicating widespread amino acid recycling via salvage pathways in conditions lacking environmental nitrogen. Our study defines an expanded quiescent proteome and phosphoproteome in yeast, and highlights the multiple coordinated molecular processes at the level of protein expression and phosphorylation that are required for quiescence.

Spontaneous single-nucleotide substitutions and microsatellite mutations have distinct distributions of fitness effects: Distributions of fitness effects of spontaneous mutations.

The fitness effects of new mutations determine key properties of evolutionary processes. Beneficial mutations drive evolution, yet selection is also shaped by the frequency of small-effect deleterious mutations, whose combined effect can burden otherwise adaptive lineages and alter evolutionary trajectories and outcomes in clonally evolving organisms such as viruses, microbes, and tumors. The small effect sizes of these important mutations have made accurate measurements of their rates difficult. In microbes, assessing the effect of mutations on growth can be especially instructive, as this complex phenotype is closely linked to fitness in clonally evolving organisms. Here, we perform high-throughput time-lapse microscopy on cells from mutation-accumulation strains to precisely infer the distribution of mutational effects on growth rate in the budding yeast, Saccharomyces cerevisiae. We show that mutational effects on growth rate are overwhelmingly negative, highly skewed towards very small effect sizes, and frequent enough to suggest that deleterious hitchhikers may impose a significant burden on evolving lineages. By using lines that accumulated mutations in either wild-type or slippage repair-defective backgrounds, we further disentangle the effects of two common types of mutations, single-nucleotide substitutions and simple sequence repeat indels, and show that they have distinct effects on yeast growth rate. Although the average effect of a simple sequence repeat mutation is very small (~0.3%), many do alter growth rate, implying that this class of frequent mutations has an important evolutionary impact.

Fluctuating Environments Maintain Genetic Diversity through Neutral Fitness Effects and Balancing Selection.

Genetic variation is the raw material upon which selection acts. The majority of environmental conditions change over time and therefore may result in variable selective effects. How temporally fluctuating environments impact the distribution of fitness effects and in turn population diversity is an unresolved question in evolutionary biology. Here, we employed continuous culturing using chemostats to establish environments that switch periodically between different nutrient limitations and compared the dynamics of selection to static conditions. We used the pooled Saccharomyces cerevisiae haploid gene deletion collection as a synthetic model for populations comprising thousands of unique genotypes. Using barcode sequencing, we find that static environments are uniquely characterized by a small number of high-fitness genotypes that rapidly dominate the population leading to dramatic decreases in genetic diversity. By contrast, fluctuating environments are enriched in genotypes with neutral fitness effects and an absence of extreme fitness genotypes contributing to the maintenance of genetic diversity. We also identified a unique class of genotypes whose frequencies oscillate sinusoidally with a period matching the environmental fluctuation. Oscillatory behavior corresponds to large differences in short-term fitness that are not observed across long timescales pointing to the importance of balancing selection in maintaining genetic diversity in fluctuating environments. Our results are consistent with a high degree of environmental specificity in the distribution of fitness effects and the combined effects of reduced and balancing selection in maintaining genetic diversity in the presence of variable selection.

Water impacts nutrient dose responses genome-wide to affect crop production.

Changes in nutrient dose have dramatic effects on gene expression and development. One outstanding question is whether organisms respond to changes in absolute nutrient amount (moles) vs. its concentration in water (molarity). This question is particularly relevant to plants, as soil drying can alter nutrient concentration, without changing its absolute amount. To compare the effects of amount vs. concentration, we expose rice to a factorial matrix varying the dose of nitrogen (N) and water (W) over a range of combinations, and quantify transcriptome and phenotype responses. Using linear models, we identify distinct dose responses to either N-moles, W-volume, N-molarity (N/W), or their synergistic interaction (N×W). Importantly, genes whose expression patterns are best explained by N-dose and W interactions (N/W or N×W) in seedlings are associated with crop outcomes in replicated field trials. Such N-by-W responsive genes may assist future efforts to develop crops resilient to increasingly arid, low nutrient soils.

A complete statistical model for calibration of RNA-seq counts using external spike-ins and maximum likelihood theory.

A fundamental assumption, common to the vast majority of high-throughput transcriptome analyses, is that the expression of most genes is unchanged among samples and that total cellular RNA remains constant. As the number of analyzed experimental systems increases however, different independent studies demonstrate that this assumption is often violated. We present a calibration method using RNA spike-ins that allows for the measurement of absolute cellular abundance of RNA molecules. We apply the method to pooled RNA from cell populations of known sizes. For each transcript, we compute a nominal abundance that can be converted to absolute by dividing by a scale factor determined in separate experiments: the yield coefficient of the transcript relative to that of a reference spike-in measured with the same protocol. The method is derived by maximum likelihood theory in the context of a complete statistical model for sequencing counts contributed by cellular RNA and spike-ins. The counts are based on a sample from a fixed number of cells to which a fixed population of spike-in molecules has been added. We illustrate and evaluate the method with applications to two global expression data sets, one from the model eukaryote Saccharomyces cerevisiae, proliferating at different growth rates, and differentiating cardiopharyngeal cell lineages in the chordate Ciona robusta. We tested the method in a technical replicate dilution study, and in a k-fold validation study.

Kinetics of Inhibitory Feedback from Horizontal Cells to Photoreceptors: Implications for an Ephaptic Mechanism.

UNLABELLED: Inhibitory feedback from horizontal cells (HCs) to cones generates center-surround receptive fields and color opponency in the retina. Mechanisms of HC feedback remain unsettled, but one hypothesis proposes that an ephaptic mechanism may alter the extracellular electrical field surrounding photoreceptor synaptic terminals, thereby altering Ca(2+) channel activity and photoreceptor output. An ephaptic voltage change produced by current flowing through open channels in the HC membrane should occur with no delay. To test for this mechanism, we measured kinetics of inhibitory feedback currents in Ambystoma tigrinum cones and rods evoked by hyperpolarizing steps applied to synaptically coupled HCs. Hyperpolarizing HCs stimulated inward feedback currents in cones that averaged 8-9 pA and exhibited a biexponential time course with time constants averaging 14-17 ms and 120-220 ms. Measurement of feedback-current kinetics was limited by three factors: (1) HC voltage-clamp speed, (2) cone voltage-clamp speed, and (3) kinetics of Ca(2+) channel activation or deactivation in the photoreceptor terminal. These factors totaled ∼4-5 ms in cones meaning that the true fast time constants for HC-to-cone feedback currents were 9-13 ms, slower than expected for ephaptic voltage changes. We also compared speed of feedback to feedforward glutamate release measured at the same cone/HC synapses and found a latency for feedback of 11-14 ms. Inhibitory feedback from HCs to rods was also significantly slower than either measurement kinetics or feedforward release. The finding that inhibitory feedback from HCs to photoreceptors involves a significant delay indicates that it is not due to previously proposed ephaptic mechanisms. SIGNIFICANCE STATEMENT: Lateral inhibitory feedback from horizontal cells (HCs) to photoreceptors creates center-surround receptive fields and color-opponent interactions. Although underlying mechanisms remain unsettled, a longstanding hypothesis proposes that feedback is due to ephaptic voltage changes that regulate photoreceptor synaptic output by altering Ca(2+) channel activity. Ephaptic processes should occur with no delay. We measured kinetics of inhibitory feedback currents evoked in photoreceptors with voltage steps applied to synaptically coupled HCs and found that feedback is too slow to be explained by ephaptic voltage changes generated by current flowing through continuously open channels in HC membranes. By eliminating the proposed ephaptic mechanism for HC feedback regulation of photoreceptor Ca(2+) channels, our data support earlier proposals that synaptic cleft pH changes are more likely responsible.

A unified framework and method for automatic neural spike identification.

Automatic identification of action potentials from one or more extracellular electrode recordings is generally achieved by clustering similar segments of the measured voltage trace, a method that fails (or requires substantial human intervention) for spikes whose waveforms overlap. We formulate the problem in terms of a simple probabilistic model, and develop a unified method to identify spike waveforms along with continuous-valued estimates of their arrival times, even in the presence of overlap. Specifically, we make use of a recent algorithm known as Continuous Basis Pursuit for solving linear inverse problems in which the component occurrences are sparse and are at arbitrary continuous-valued times. We demonstrate significant performance improvements over current state-of-the-art clustering methods for four simulated and two real data sets with ground truth, each of which has previously been used as a benchmark for spike sorting. In addition, performance of our method on each of these data sets surpasses that of the best possible clustering method (i.e., one that is specifically optimized to minimize errors on each data set). Finally, the algorithm is almost completely automated, with a computational cost that scales well for multi-electrode arrays.

An alternating renewal process describes the buildup of perceptual segregation.

For some ambiguous scenes perceptual conflict arises between integration and segregation. Initially, all stimulus features seem integrated. Then abruptly, perhaps after a few seconds, a segregated percept emerges. For example, segregation of acoustic features into streams may require several seconds. In behavioral experiments, when a subject's reports of stream segregation are averaged over repeated trials, one obtains a buildup function, a smooth time course for segregation probability. The buildup function has been said to reflect an underlying mechanism of evidence accumulation or adaptation. During long duration stimuli perception may alternate between integration and segregation. We present a statistical model based on an alternating renewal process (ARP) that generates buildup functions without an accumulative process. In our model, perception alternates during a trial between different groupings, as in perceptual bistability, with random and independent dominance durations sampled from different percept-specific probability distributions. Using this theory, we describe the short-term dynamics of buildup observed on short trials in terms of the long-term statistics of percept durations for the two alternating perceptual organizations. Our statistical-dynamics model describes well the buildup functions and alternations in simulations of pseudo-mechanistic neuronal network models with percept-selective populations competing through mutual inhibition. Even though the competition model can show history dependence through slow adaptation, our statistical switching model, that neglects history, predicts well the buildup function. We propose that accumulation is not a necessary feature to produce buildup. Generally, if alternations between two states exhibit independent durations with stationary statistics then the associated buildup function can be described by the statistical dynamics of an ARP.

Plasticity regulators modulate specific root traits in discrete nitrogen environments.

Plant development is remarkably plastic but how precisely can the plant customize its form to specific environments? When the plant adjusts its development to different environments, related traits can change in a coordinated fashion, such that two traits co-vary across many genotypes. Alternatively, traits can vary independently, such that a change in one trait has little predictive value for the change in a second trait. To characterize such "tunability" in developmental plasticity, we carried out a detailed phenotypic characterization of complex root traits among 96 accessions of the model Arabidopsis thaliana in two nitrogen environments. The results revealed a surprising level of independence in the control of traits to environment - a highly tunable form of plasticity. We mapped genetic architecture of plasticity using genome-wide association studies and further used gene expression analysis to narrow down gene candidates in mapped regions. Mutants in genes implicated by association and expression analysis showed precise defects in the predicted traits in the predicted environment, corroborating the independent control of plasticity traits. The overall results suggest that there is a pool of genetic variability in plants that controls traits in specific environments, with opportunity to tune crop plants to a given environment.

Modulation of mouse rod response decay by rhodopsin kinase and recoverin.

Light isomerizes 11-cis-retinal in a retinal rod and produces an active form of rhodopsin (Rh*) that binds to the G-protein transducin and activates the phototransduction cascade. Rh* is turned off by phosphorylation by rhodopsin kinase [G-protein-coupled receptor kinase 1 (GRK1)] and subsequent binding of arrestin. To evaluate the role of GRK1 in rod light response decay, we have generated the transgenic mouse RKS561L in which GRK1, which is normally present at only 2-3% of rhodopsin, is overexpressed by ∼12-fold. Overexpression of GRK1 increases the rate of Rh* phosphorylation and reduces the exponential decay constant of the response (τ(REC)) and the limiting time constant (τ(D)) both by ∼30%; these decreases are highly significant. Similar decreases are produced in Rv(-/-) rods, in which the GRK1-binding protein recoverin has been genetically deleted. These changes in response decay are produced by acceleration of light-activated phosphodiesterase (PDE*) decay rather than Rh* decay, because light-activated PDE* decay remains rate limiting for response decay in both RKS561L and Rv(-/-) rods. A model incorporating an effect of GRK1 on light-activated PDE* decay rate can satisfactorily account for the changes in response amplitude and waveform. Modulation of response decay in background light is nearly eliminated by deletion of recoverin. Our experiments indicate that rhodopsin kinase and recoverin, in addition to their well-known role in regulating the turning off of Rh*, can also modulate the decay of light-activated PDE*, and the effects of these proteins on light-activated PDE* decay may be responsible for the quickening of response recovery in background light.

Population density approach for discrete mRNA distributions in generalized switching models for stochastic gene expression.

We present a generalization of a population density approach for modeling and analysis of stochastic gene expression. In the model, the gene of interest fluctuates stochastically between an inactive state, in which transcription cannot occur, and an active state, in which discrete transcription events occur; and the individual mRNA molecules are degraded stochastically in an independent manner. This sort of model in simplest form with exponential dwell times has been used to explain experimental estimates of the discrete distribution of random mRNA copy number. In our generalization, the random dwell times in the inactive and active states, T_{0} and T_{1}, respectively, are independent random variables drawn from any specified distributions. Consequently, the probability per unit time of switching out of a state depends on the time since entering that state. Our method exploits a connection between the fully discrete random process and a related continuous process. We present numerical methods for computing steady-state mRNA distributions and an analytical derivation of the mRNA autocovariance function. We find that empirical estimates of the steady-state mRNA probability mass function from Monte Carlo simulations of laboratory data do not allow one to distinguish between underlying models with exponential and nonexponential dwell times in some relevant parameter regimes. However, in these parameter regimes and where the autocovariance function has negative lobes, the autocovariance function disambiguates the two types of models. Our results strongly suggest that temporal data beyond the autocovariance function is required in general to characterize gene switching.

Recovery of sparse translation-invariant signals with continuous basis pursuit.

We consider the problem of decomposing a signal into a linear combination of features, each a continuously translated version of one of a small set of elementary features. Although these constituents are drawn from a continuous family, most current signal decomposition methods rely on a finite dictionary of discrete examples selected from this family (e.g., shifted copies of a set of basic waveforms), and apply sparse optimization methods to select and solve for the relevant coefficients. Here, we generate a dictionary that includes auxiliary interpolation functions that approximate translates of features via adjustment of their coefficients. We formulate a constrained convex optimization problem, in which the full set of dictionary coefficients represents a linear approximation of the signal, the auxiliary coefficients are constrained so as to only represent translated features, and sparsity is imposed on the primary coefficients using an L1 penalty. The basis pursuit denoising (BP) method may be seen as a special case, in which the auxiliary interpolation functions are omitted, and we thus refer to our methodology as continuous basis pursuit (CBP). We develop two implementations of CBP for a one-dimensional translation-invariant source, one using a first-order Taylor approximation, and another using a form of trigonometric spline. We examine the tradeoff between sparsity and signal reconstruction accuracy in these methods, demonstrating empirically that trigonometric CBP substantially outperforms Taylor CBP, which in turn offers substantial gains over ordinary BP. In addition, the CBP bases can generally achieve equally good or better approximations with much coarser sampling than BP, leading to a reduction in dictionary dimensionality.

Channel modulation and the mechanism of light adaptation in mouse rods.

Vertebrate photoreceptors are thought to adapt to light by a change in Ca(2+), which is postulated to mediate modulation of (1) excited rhodopsin (Rh*) by Ca(2+)-dependent binding of recoverin, (2) guanylyl cyclase activity via Ca(2+)-dependent GCAP proteins, and (3) cyclic nucleotide-gated channels by binding of Ca(2+)-calmodulin. Previous experiments genetically deleted recoverin and the GCAPs and showed that significant regulation of sensitivity survives removal of (1) and (2). We genetically deleted the channel Ca(2+)-calmodulin binding site in the mouse Mus musculus and found that removal of (3) alters response waveform, but removal of (3) or of (2) and (3) together still leaves much of adaptation intact. These experiments demonstrate that an important additional mechanism is required, which other experiments indicate may be regulation of phosphodiesterase 6 (PDE6). We therefore constructed a kinetic model in which light produces a Ca(2+)-mediated decrease in PDE6 decay rate, with the novel feature that both spontaneously activated and light-activated PDE6 are modulated. This model, together with Ca(2+)-dependent acceleration of guanylyl cyclase, can successfully account for changes in sensitivity and response waveform in background light.

In silico evaluation of predicted regulatory interactions in Arabidopsis thaliana.

BACKGROUND: Prediction of transcriptional regulatory mechanisms in Arabidopsis has become increasingly critical with the explosion of genomic data now available for both gene expression and gene sequence composition. We have shown in previous work 1, that a combination of correlation measurements and cis-regulatory element (CRE) detection methods are effective in predicting targets for candidate transcription factors for specific case studies which were validated. However, to date there has been no quantitative assessment as to which correlation measures or CRE detection methods used alone or in combination are most effective in predicting TF-->target relationships on a genome-wide scale. RESULTS: We tested several widely used methods, based on correlation (Pearson and Spearman Rank correlation) and cis-regulatory element (CRE) detection (>or=1 CRE or CRE over-representation), to determine which of these methods individually or in combination is the most effective by various measures for making regulatory predictions. To predict the regulatory targets of a transcription factor (TF) of interest, we applied these methods to microarray expression data for genes that were regulated over treatment and control conditions in wild type (WT) plants. Because the chosen data sets included identical experimental conditions used on TF over-expressor or T-DNA knockout plants, we were able to test the TF-->target predictions made using microarray data from WT plants, with microarray data from mutant/transgenic plants. For each method, or combination of methods, we computed sensitivity, specificity, positive and negative predictive value and the F-measure of balance between sensitivity and positive predictive value (precision). This analysis revealed that the >or=1 CRE and Spearman correlation (used alone or in combination) were the most balanced CRE detection and correlation methods, respectively with regard to their power to accurately predict regulatory-target interactions. CONCLUSION: These findings provide an approach and guidance for researchers interested in predicting transcriptional regulatory mechanisms using microarray data that they generate (or microarray data that is publically available) combined with CRE detection in promoter sequence data.

A system biology approach highlights a hormonal enhancer effect on regulation of genes in a nitrate responsive "biomodule".

BACKGROUND: Nitrate-induced reprogramming of the transcriptome has recently been shown to be highly context dependent. Herein, a systems biology approach was developed to identify the components and role of cross-talk between nitrate and hormone signals, likely to be involved in the conditional response of NO3- signaling. RESULTS: Biclustering was used to identify a set of genes that are N-responsive across a range of Nitrogen (N)-treatment backgrounds (i.e. nitrogen treatments under different growth conditions) using a meta-dataset of 76 Affymetrix ATH1 chips from 5 different laboratories. Twenty-one biclusters were found to be N-responsive across subsets of this meta-dataset. N-bicluster 9 (126 genes) was selected for further analysis, as it was shown to be reproducibly responsive to NO3- as a signal, across a wide-variety of background conditions and datasets. N-bicluster 9 genes were then used as "seed" to identify putative cross-talk mechanisms between nitrate and hormone signaling. For this, the 126 nitrate-regulated genes in N-bicluster 9 were biclustered over a meta-dataset of 278 ATH1 chips spanning a variety of hormone treatments. This analysis divided the bicluster 9 genes into two classes: i) genes controlled by NO3- only vs. ii) genes controlled by both NO3- and hormones. The genes in the latter group showed a NO3- response that is significantly enhanced, compared to the former. In silico analysis identified two Cis-Regulatory Elements candidates (CRE) (E2F, HSE) potentially involved the interplay between NO3- and hormonal signals. CONCLUSION: This systems analysis enabled us to derive a hypothesis in which hormone signals are proposed to enhance the nitrate response, providing a potential mechanistic explanation for the link between nitrate signaling and the control of plant development.

Spike train statistics and dynamics with synaptic input from any renewal process: a population density approach.

In the probability density function (PDF) approach to neural network modeling, a common simplifying assumption is that the arrival times of elementary postsynaptic events are governed by a Poisson process. This assumption ignores temporal correlations in the input that sometimes have important physiological consequences. We extend PDF methods to models with synaptic event times governed by any modulated renewal process. We focus on the integrate-and-fire neuron with instantaneous synaptic kinetics and a random elementary excitatory postsynaptic potential (EPSP), A. Between presynaptic events, the membrane voltage, v, decays exponentially toward rest, while s, the time since the last synaptic input event, evolves with unit velocity. When a synaptic event arrives, v jumps by A, and s is reset to zero. If v crosses the threshold voltage, an action potential occurs, and v is reset to v(reset). The probability per unit time of a synaptic event at time t, given the elapsed time s since the last event, h(s, t), depends on specifics of the renewal process. We study how regularity of the train of synaptic input events affects output spike rate, PDF and coefficient of variation (CV) of the interspike interval, and the autocorrelation function of the output spike train. In the limit of a deterministic, clocklike train of input events, the PDF of the interspike interval converges to a sum of delta functions, with coefficients determined by the PDF for A. The limiting autocorrelation function of the output spike train is a sum of delta functions whose coefficients fall under a damped oscillatory envelope. When the EPSP CV, sigma A/mu A, is equal to 0.45, a CV for the intersynaptic event interval, sigma T/mu T = 0.35, is functionally equivalent to a deterministic periodic train of synaptic input events (CV = 0) with respect to spike statistics. We discuss the relevance to neural network simulations.

A systems approach uncovers restrictions for signal interactions regulating genome-wide responses to nutritional cues in Arabidopsis.

As sessile organisms, plants must cope with multiple and combined variations of signals in their environment. However, very few reports have studied the genome-wide effects of systematic signal combinations on gene expression. Here, we evaluate a high level of signal integration, by modeling genome-wide expression patterns under a factorial combination of carbon (C), light (L), and nitrogen (N) as binary factors in two organs (O), roots and leaves. Signal management is different between C, N, and L and in shoots and roots. For example, L is the major factor controlling gene expression in leaves. However, in roots there is no obvious prominent signal, and signal interaction is stronger. The major signal interaction events detected genome wide in Arabidopsis roots are deciphered and summarized in a comprehensive conceptual model. Surprisingly, global analysis of gene expression in response to C, N, L, and O revealed that the number of genes controlled by a signal is proportional to the magnitude of the gene expression changes elicited by the signal. These results uncovered a strong constraining structure in plant cell signaling pathways, which prompted us to propose the existence of a "code" of signal integration.

Critical analysis of dimension reduction by a moment closure method in a population density approach to neural network modeling.

Computational techniques within the population density function (PDF) framework have provided time-saving alternatives to classical Monte Carlo simulations of neural network activity. Efficiency of the PDF method is lost as the underlying neuron model is made more realistic and the number of state variables increases. In a detailed theoretical and computational study, we elucidate strengths and weaknesses of dimension reduction by a particular moment closure method (Cai, Tao, Shelley, & McLaughlin, 2004; Cai, Tao, Rangan, & McLaughlin, 2006) as applied to integrate-and-fire neurons that receive excitatory synaptic input only. When the unitary postsynaptic conductance event has a single-exponential time course, the evolution equation for the PDF is a partial differential integral equation in two state variables, voltage and excitatory conductance. In the moment closure method, one approximates the conditional kth centered moment of excitatory conductance given voltage by the corresponding unconditioned moment. The result is a system of k coupled partial differential equations with one state variable, voltage, and k coupled ordinary differential equations. Moment closure at k = 2 works well, and at k = 3 works even better, in the regime of high dynamically varying synaptic input rates. Both closures break down at lower synaptic input rates. Phase-plane analysis of the k = 2 problem with typical parameters proves, and reveals why, no steady-state solutions exist below a synaptic input rate that gives a firing rate of 59 s(1) in the full 2D problem. Closure at k = 3 fails for similar reasons. Low firing-rate solutions can be obtained only with parameters for the amplitude or kinetics (or both) of the unitary postsynaptic conductance event that are on the edge of the physiological range. We conclude that this dimension-reduction method gives ill-posed problems for a wide range of physiological parameters, and we suggest future directions.

Population density methods for stochastic neurons with realistic synaptic kinetics: firing rate dynamics and fast computational methods.

An outstanding problem in computational neuroscience is how to use population density function (PDF) methods to model neural networks with realistic synaptic kinetics in a computationally efficient manner. We explore an application of two-dimensional (2-D) PDF methods to simulating electrical activity in networks of excitatory integrate-and-fire neurons. We formulate a pair of coupled partial differential-integral equations describing the evolution of PDFs for neurons in non-refractory and refractory pools. The population firing rate is given by the total flux of probability across the threshold voltage. We use an operator-splitting method to reduce computation time. We report on speed and accuracy of PDF results and compare them to those from direct, Monte-Carlo simulations. We compute temporal frequency response functions for the transduction from the rate of postsynaptic input to population firing rate, and examine its dependence on background synaptic input rate. The behaviors in the1-D and 2-D cases--corresponding to instantaneous and non-instantaneous synaptic kinetics, respectively--differ markedly from those for a somewhat different transduction: from injected current input to population firing rate output (Brunel et al. 2001; Fourcaud & Brunel 2002). We extend our method by adding inhibitory input, consider a 3-D to 2-D dimension reduction method, demonstrate its limitations, and suggest directions for future study.