Workshop 5: Cellular and Subcellular (April 8-12, 2013)
Organizers: Janet Best, Avrama Blackwell, and Paul Bressloff
Mathematical modeling has become a widely-used tool for integrating biological data, designing experiments, and ultimately understanding biological systems. In recent years two important challenges for the successful use of mathematical models have become apparent. One is that models contain parameters that determine the behavior of the model, and the values of these parameters are often hard to determine from the available biological data. The other challenge is that many biological systems exhibit a great deal of heterogeneity in behavior, so even if the model parameters could be perfectly calibrated by pooling cell behaviors to produce an “average cell model”, this model may not provide a good description of any single cell in the population. I will describe some of the techniques that we are using to integrate mathematical modeling into experimental studies in a way that addresses both of these challenges. We study endocrine pituitary cells that release a variety of hormones into the blood, and our aim is to develop an approach for modeling the behaviors of these cells with enough accuracy so that in spite of heterogeneity within the cell population we can use the models to make, and subsequently test, predictions.
Calcium-calmodulin-dependent protein kinase II (CaMKII) is a key regulator of glutamatergic synapses and plays an essential role in many forms of synaptic plasticity. It has recently been observed experimentally that stimulating a local region of dendrite not only induces the local translocation of CaMKII from the dendritic shaft to synaptic targets within spines, but also initiates a wave of CaMKII translocation that spreads distally along the dendrite with an average speed of order 1 micron/s. In this talk we present a simple reaction--diffusion model of CaMKII translocation waves that can account for the observed wave speed and predicts wave propagation failure if the density of spines is too high. We also analyze how heterogeneities in the spine distribution modulate the speed of CaMKII translocation waves. These heterogeneities occur on two different spatial scales. First, spines are discrete entities that are joined to a dendritic branch via a thin spine neck of submicron radius, resulting in spatial variations in spine density at the micron level. The second source of heterogeneity occurs on a much longer length scale and reflects the experimental observation that there is a slow proximal to distal variation in the density of spines. We adapt methods from the study of the spread of biological invasions in heterogeneous environments, including homogenization theory of pulsating fronts and Hamilton-Jacobi dynamics of sharp interfaces.
Calcium is known to play a fundamental role in synaptic plasticity. However, it is still unclear to what extent the dynamics of calcium concentration in post-synaptic spines alone can account for the phenomenology of plasticity. In this talk, I will first present a simplified calcium-based synaptic plasticity model, and show that it can reproduce quantitatively a large amount of experimental data in several preparations (hippocampal cultures, hippocampal slices and cortical slices). Differences between plasticity outcomes in such preparations are predicted to arise due to differences in parameters controlling calcium dynamics (such as the extracellular calcium concentration). I will then present some consequences of this plasticity rule at the network level.
Dopamine neurons in freely moving rats often fire behaviorally-relevant high frequency bursts, but depolarization block limits the maximum steady firing rate of dopamine neurons in vitro to approximately 10 Hz. Using a reduced model that faithfully reproduces the sodium current measured in these neurons, we show that adding an additional slow component of sodium channel inactivation, recently observed in these neurons, qualitatively changes in two different ways how the model enters depolarization block. First, the slow time course of inactivation allows multiple spikes with progressively increasing interspike intervals to be elicited during a strong depolarization prior to entry into depolarization block, which may be critical for the ability to burst in vivo. Second, depolarization block occurs much closer to spike threshold, because the additional slow component of inactivation negates the sodium window current. In the absence of the additional slow component of inactivation, this window current produces an S-shaped steady state IV curve that prevents depolarization block in the experimentally observed voltage range near -40 mV. Significantly, the time constant of recovery from slow inactivation during the interspike interval limits the maximum steady firing rate observed prior to entry into depolarization block. These qualitative features of the entry into depolarization block can be reversed experimentally by replacing the native sodium conductance with a virtual one lacking the slow component of inactivation. Our modeling results also suggest that activation of NMDA receptors may contribute to circumventing the firing rate limitation during behaviorally relevant, high frequency bursts in vivo.
The spatial component of input signals often carries information crucial to a neuron's function, but models which map synaptic inputs to transmembrane potential can be computationally expensive. Existing reduced models of the neuron either merge compartments, thereby sacrificing the spatial specificity of inputs, or apply model reduction techniques which sacrifice the biological interpretation of the model. We use Krylov subspace projection methods to construct reduced models of the quasi-active neurons which preserve both the spatial specificity of inputs and the biological interpretation as an RLC circuit, respectively. Each reduced model accurately computes the potential at the spike initiation zone given a much smaller dimension and simulation time, as we show numerically and theoretically. The structure is preserved through the similarity in the circuit representations, for which we provide circuit diagrams and mathematical expressions for the circuit elements. Furthermore, the transformation from the full to the reduced system is straightforward and depends on the intrinsic properties of the dendrite. As each reduced model is accurate and has a clear biological interpretation, the reduced models can be used not only to simulate morphologically accurate neurons but also to examine the underlying functions performed in dendrites.
The neuropeptide orexin/hypocretin is essential for normal consolidation of sleep/wake behavior, and disruption of the orexin system is associated with the sleep disorder narcolepsy. Recent experimental work has characterized elements of orexin neuron electrophysiology and state-dependent behavior, however, many questions, particularly questions of dynamics, can be difficult to address in an experimental setting. I will discuss several modeling approaches, spanning multiple scales, which we have undertaken to investigate the intrinsic dynamics of these neurons and their role in sleep/wake regulation.
A local field potential is measured by an electrode inserted into a certain region in the brain. Data show bursts of activity at a number of frequencies, including the gamma region of 40-100Hz. Here we study, with mathematical analysis, and supporting simulation data, a simple, linear stochastic model of the generation of gamma bursts in local field potential (LFP) recordings by interacting populations of excitatory and inhibitory neurons. By analyzing this simple model, which represents a large class of similar but more complicated models, we are able to proceed further with mathematical analysis than has been accomplished previously. We show that the simple stochastic model can be approximated in terms of a rotation multiplied by a two-dimensional Ornstein-Uhlenbeck (OU) process. We demonstrate that gamma bursts arise in the model as excursions of the modulus of the OU process. Finally, there is a reciprocal relationship between the amplitude of the envelope of the gamma oscillation and the time derivative of phase that, among other properties of the approximation, is mirrored in LFP data simulated from the original model. The close relationship between the properties of the approximation and those of simulations of the original model suggests that the approximation is a valid representation for a wide class of models of oscillatory neural processes.
What defines synaptic strength at a molecular level and how can we compute the synaptic current? To answer these questions, we will present mathematical models that we have developed for estimating the current at excitatory synapses based on the properties of AMPA receptors. We accounted for various geometrical parameters of the synapse and also for receptor trafficking. We will also discuss statistical methods based on the Langevin's equation to extract local biophysical properties of cell-particle interaction from thousands of individual trajectories. We will focus on AMPA receptor diffusion properties and the strength of their molecular interaction at the sub-diffraction level. The present analysis reveals several attracting potential wells of large sizes, showing that the high density of AMPARs is generated by physical interactions with an ensemble of cooperative membrane surface binding sites, rather than molecular crowding. Moreover, potential wells control the flux of receptors at the base of dendritic spines.
This talk summarizes our long lasting effort to identify key parameters involved in the regulation of synaptic transmission and plasticity, processes that underlie learning and memory.
Excitatory synapses in pyramidal neurons are distributed on spines spread over extensively arborized dendrites. These inputs are the sites of contact for a large fraction of the excitatory synapses in the mammalian brain, and as a result such dendritic inputs are the first step in the signaling between such inputs and a neuron’s action potential output. In a combined computational and experimental study of CA1 pyramidal neurons, we demonstrate how spatially varying distributions of synapse number and size combine to influence somatic membrane potential and action potential initiation in the axon, which often can be hundreds of microns away from the site of the inputs. We also demonstrate that spines provide a uniformly high impedance compartment across the dendritic arbor that amplifies local depolarization. This spine amplification increases nonlinear voltage-dependent conductance activation and promotes electrical interaction among coactive inputs, enhancing neuronal response.
Information is stored in the brain through the formation of neural networks that encode memories. New networks are formed when the strength of synapses connecting groups of neurons increases. To achieve accurate and efficient storage of information in the brain, synaptic plasticity in the cortex and hippocampus is delicately regulated by the patterns of activity at each synapse. Ca2+ influx through NMDA-type glutamate receptors triggers the biochemical processes that lead to either long-term potentiation (LTP) of the strength of the synapse, or long-term depression (LTD). We still do not understand how a small change in the rate and extent of flux of Ca2+ into the spine can bring about a large change in the nature of the alteration of the structure of the spine and the strength of the synapse.
Understanding the molecular processes that govern synaptic strength is important for our understanding of brain function as a whole; however, it is especially important in the context of mental illness. Mutation of proteins that control synaptic plasticity, or that tune the dynamics of biochemistry in the spine by acting as scaffolds, produces increased risk for the development of mental illnesses such as schizophrenia, autism, and bipolar disease, and for certain forms of mental retardation.
I will discuss how we are applying computational methods and computer modeling to aid our understanding of the dynamics of enzyme regulation by Ca2+ in the spine. We use a well-established agent-based, stochastic modeling program called MCell. The nature of signaling machinery inside the spine requires “agent-based” modeling. The program MCell and the open-source model-building tool Blender, provide a powerful system for constructing and visualizing such models. I will present early results from our modeling efforts in collaboration with Tom Bartol of the Sejnowski laboratory at the Salk Institute, and Kristen Harris and Chandrajit Bajaj at University of Texas, Austin.
The NEURON simulator is a widely used tool for studying detailed single cell and network models. In recognition of the growing importance of multiscale modeling, we have expanded NEURON’s support for intracellular chemical dynamics. Our initial work has explored deterministic reactiondiffusion models with onedimensional simulation. Unlike previous NEURON mechanisms, arbitrary new reaction schemes may be specified at runtime via HOC or Python; no separate compilation step is required. This flexibility will allow us to import models written in the Systems Biology and in Virtual Cell Markup Languages (SBML, VCML), which will facilitate collaboration between the neuroscience and cell biology communities. In certain situations, such as calcium dynamics near a spine, only a few particles of a given chemical species are present. As these particles randomly move around, there is the potential for large percentage deviations from the mean concentration. To study these effects, we will support Gillespie and tauleaping algorithms for stochastic reactiondiffusion, to be designed so as to interface with deterministic diffusion.
As we moved from 1D to 3D spatial simulation, we noted additional problems in the use of standard neural morphologies. These morphologies (from Neurolucida or similar tracing systems) save cell details in a reduced format that does not fully define the surface of a cell beyond the soma. Therefore, our first step has been to define the shape of the joins between dendritic sections. We have provided this reconstruction in a way that allows the surface tesselation to be readily matched with an internal cubic mesh. We have used these tool to develop neuronal calcium (Ca2+) simulations in dendrite as a testsuite. Ca2+ waves interact bidirectionally with electrical activity. We tested different distributions and densities of IP3 receptors in the dendrite, assessing the effects on speed and strength of Ca2+ wave boosting.
A pivotal question in cell biology is whether axons elongate by the assembly of new material or bulk advance of the growth cone. While classic studies suggest the former, recent work suggests forces drive translocation of the growth cone. Here we ask three questions: Is the mechanism of growth cone advance conserved between vertebrate and invertebrate neurons? How do growth cones ad- vance in vivo? And, what is the role of myosin II force generation :in growth cone motility? To address conservation, we analyzed the movement of organelles and microtubules in neurons cultured from Drosophila embryos. We found these moved in bulk as observed in chick sensory and Xenopus spinal cord neurons. To assess transport in vivo, we co-expressed myr-td-tomato and mito-GFP in stage 16 Drosophila embryos using the pan neuronal driver elav. using time-lapse confocal microscopy to track the elongation of the aCC pioneer motor axon in intact embryos, we also found bulk advance of docked mitochondria. To better understand the role of myosin II in axonal elon- gation, we cultured Drosophila neurons that were null for myosin II (Zipper) and monitored bulk transport and growth cone motility. We found both rates were significantly higher. using force calibrated towing needles, we found dis- ruption of myosin II significantly decreased neuronal tension. Together, this suggests axonal myosin II acts antagonistically against forces generated in the growth cone to modulate translocation of the growth cone. This work has important implications for the development of treatments for stroke, peripheral nerve damage, and spinal cord injury.
A key component in the cellular mechanisms underlying learning and memory involves the distribution and delivery of mRNA to synaptic sites in dendrites. A minimal three-state random intermittent search model of motor-driven mRNA transport is developed to explore the question of why motor-driven mRNA are observed moving bidirectionally. The model is analyzed by computing the probability an mRNA is delivered to a synaptic target and the average delivery time (MFPT). It is found that if the branched geometry of the dendrite is ignored, a purely unidirectional transport strategy will result in the smallest MFPT at any given delivery probability. The branched geometry of the dendrite is then incorporated into the model, and it is shown that a phase transition exists for a critical delivery probability where bidirectional strategies improve the corresponding MFPT. To further explore the impact of motor-driven transport behavior on mRNA delivery, the three-state model is extended to include a detailed, biophysical model of a multimotor complex coordinated through a tug-of-war. The model is analyzed to explore how various measurable, physical quantities, such as adenosine triphosphate, can be tuned to optimize cargo delivery.
The cumulative effects of a wide range of heterogeneous components found in cells and networks at multiple scales could give rise to reaction-diffusion processes away from equilibrium. This complex behavior can result in the breakdown of classical laws of reaction diffusion which could give rise to power-law distributions. I will present our combined experimental and computational work that shows the breakdown of classic diffusion at multiple scales in single neurons. I will start by showing that molecular crowding in the post-synaptic density causes anomalous diffusion of glutamate receptors. This process is able to explain the results from single particle tracking experiments and provides a low energy strategy to retain glutamate receptors in the synapse for long periods of time. At a spatial scale two order of magnitude larger than a synapse I will show that the presence of dendritic spines causes anomalous diffusion of soluble cytosolic signals. This type of anomalous diffusion affects the integration of second messengers involved in synaptic plasticity. Our recent simulation on chloride dynamics predict that this important ion also undergoes anomalous diffusion in spiny dendrites. I will then present our efforts to generalize the analysis of reaction-diffusion systems outside equilibrium by using fractional reaction diffusion equations. I will explain how we are using fractional dynamics not only to study biochemical integration in dendritic trees but also how this can be used to study other types of power-law dynamics in neuronal activity such as in slowly adapting spiking trains.
Memories are stored via changes in concentrations or states of specific molecules in synapses. A central question in learning and memory is how memories can be stored for time periods that are much longer than the lifetime of these molecules in the synapse. There is significant evidence that the formation of long term memory is correlated with persistent increase of a specific kinase, PKMζ, and that inactivating this molecule can reverse previously established synaptic plasticity and memory. We construct severalmodels explaining how PKMζ can be persistent and active for periods of time larger than the protein’s lifetime. We base these models on experimental observations, and add complexities only when necessary to account for the data. Doing this we construct a model that can sufficiently and qualitatively account for most experimental data, yet is simple, tractable and can be fully mathematically analyzed. Thus we identify key characteristics of a protein necessary for maintaining the life of a memory, advancing our current understanding of how memories last.
Although it is known that inhibitory cells or interneurons represent a minority (<20%) of neurons in the brain, and that there is a wide diversity in the properties of these cells, it is unclear how these diverse cells contribute to functional output.
In this talk, I will describe our work regarding the development and use of a class of interneuron models in hippocampus that express subthreshold theta oscillations. We develop a biophysically-based cellular model and go on to use the model to examine how it could contribute to population theta rhythms. We use a computational approach in which we determine what in vivo-like conditions might support the reliable firing of these cells at theta frequencies. We find that noisy inhibitory inputs promote this and that biophysical properties that contribute to reliable firing differ from those contributing to subthreshold activities. This work thus shows how hippocampal cellular details could support functional output.
Localized Ca elevations known as Ca puffs and sparks are cellular signals that arise from the cooperative activity of clusters of inositol 1,4,5-trisphosphate receptors and ryanodine receptors clustered at Ca release sites on the surface of the endoplasmic reticulum or sarcoplasmic reticulum. When Markov chain models of intracellular Ca regulated Ca channels are coupled via a mathematical representation of a Ca microdomain, simulated Ca release sites may exhibit the phenomenon of stochastic Ca excitability where the IP3Rs or RyRs open and close in a concerted fashion. Such mathematical models provide insight into the relationship between single-channel kinetics and the statistics of puff/spark duration, and clarify the role of stochastic attrition, Ca inactivation, luminal depletion, and allosteric interactions in the dynamics of puff/spark termination. The stochastic dynamics of local Ca is an important aspect of excitation-contraction coupling in cardiac myocytes, where sarcoplasmic reticulum Ca-induced Ca release is locally controlled by trigger Ca influx via L-type channels of the plasma membrane. A recently developed whole cell modeling approach is able to avoid the computationally demanding task of resolving spatial aspects of global Ca signaling by using probability densities and associated moment equations to representing heterogeneous local Ca signals in a population of Ca release units. This new class of whole cell models of Ca handling facilitates simulation and analysis of the bidirectional coupling of localized calcium elevations and whole cell calcium responses in cardiac myocytes.
Rhythmic processes as diverse as heartbeat, respiration, locomotion, and feeding face a common challenge: adaptive regulation of central pattern generator (CPG) activity in response to varying operational demands. Mathematical models of CPGs typically involve nonlinear limit cycle (LC) dynamics. Empirically, many rhythmic processes appear to decompose into sequences of LC components, transitions between which are controlled by sensory feedback from the system external to the CPG circuit. Checkpoint based control of the budding yeast cell cycle provides an example at the cellular regulatory level. We will discuss two other examples: respiratory control, and control of a feeding pattern generator in the sea slug Aplysia californica. In the first example, we consider the behavior of a conductance-based respiratory pacemaker cell model, the Butera-Rinzel-Smith (BRS) conditional bursting neuron, when embedded in a closed-loop respiratory control model. We show that the closed-loop model is bistable, with one stable state (bursting activity) corresponding to "eupnea" or normal breathing, and a second stable state (beating activity) that may correspond to "tachypnea", or pathological rapid shallow breathing. The BRS conductances also show a novel "autoresuscitation" effect, in which a transient drop of blood oxygen content causes a large burst of inspiratory activity.
In the second part of the talk, we will consider a nominal model for a motor control problem: protraction and retraction of the mouthparts of Aplysia while feeding on seaweed. By varying a parameter, the underlying LC driving the feeding pattern generator can be moved between a Hopf bifurcation and a heteroclinic bifurcation. We ask how the model responds to natural perturbations by adding an external load to mimic the resistance of seaweed to being swallowed during feeding. When the LC is close to the heteroclinic bifurcation, the musculature responds by prolonging the retraction phase of the motion. When the LC is close to the Hopf bifurcation, the musculature response is less adaptive to the load. If the role of sensory feedback is to permit transit from one phase of an iterated behavior to the next, then in the absence of sensory feedback we would predict uniform prolongation of all phases of the motion. Experimental results from the Chiel laboratory suggest that sensory feedback guides the timing of the feeding pattern generator by steering trajectories closer to or farther from a succession of fixed points.
Joint work with Hillel J. Chiel, Miranda J. Cullins, Casey O. Diekman, Hui Lu, Jeffrey M. McManus, Youngmin Park, Kendrick M. Shaw, Christopher G. Wilson
Gap junctions, also referred to as electrical synapses, are expressed along the entire central nervous system and are important in mediating various brain rhythms in both normal and pathological states. These connections can form between the dendritic trees of individual cells. Many dendrites express membrane channels that confer on them a form of sub-threshold resonant dynamics. To obtain insight into the modulatory role of gap junctions in tuning networks of resonant dendritic trees we generalise the "sum-over-trips" formalism to treat networks of dendritic trees connected via dendro-dendritic gap junctions. Applying this framework to a two-cell network we construct compact closed form solutions for the network response function in the Laplace (frequency) domain and study how a preferred frequency in each soma depends on the location and strength of the gap junction.
The variability of the postsynaptic response following a single action potential arises from two sources: the neurotransmitter release is probabilistic, and the postsynaptic response to neurotransmitter release has variable timing and amplitude. At individual synapses, the number of molecules of a given type that are involved in these processes is small enough that the stochastic (random) properties of molecular events cannot be neglected. How the stochasticity of molecular processes contributes to the variability of synaptic transmission, its sensitivity and its robustness to molecular fluctuations has important implications for our understanding of the mechanistic basis of synaptic transmission and of synaptic plasticity. Using single particle tracking and super-resolution imaging, we will address the issue of postsynaptic receptors dynamic, their interactions with scaffolding protein and regulations implicated in synaptic plasticity. Combination of single particle tracking and super-resolution methods, open access to molecular counting and energy involved in receptor-scaffold interactions as well as on and off rate of molecular interactions. Thus beyond super-resolution methods is chemistry “in cellulo” accounting for the regulation of receptor number and consecutively that of synaptic strength.
Poster
Calcium (Ca2+) waves provide a complement to electrical signaling in the neuron, forming a key part of a neuron's second messenger system. This chemical phenomenon interacts bidirectionally with electrical activity. Ca2+ is normally sequestered into the endoplasmic reticulum (ER) via pumping of the sarco/endoplasmic reticulum Ca2+ ATPase (SERCA). Inositol triphosphate (IP3) is a ligand that regulates admittance of Ca2+ from ER into the cytosol when it binds to IP3 receptors (IP3R) on the ER. ER IP3Rs are also co-regulated by Ca2+. Therefore, when Ca2+ binds to IP3Rs, it can induce its own release into the cytosol (calcium-induced calcium release: CICR). These phenomena have clinical relevance: dysregulation of Ca2+ channels of the smooth endoplasmic reticulum (SER) has been implicated in Alzheimer's and Huntington's disease. We developed a reaction-diffusion model of a neuron which contained separate cytosolic and ER volumes, each containing their own local Ca2+ concentrations. Our model included diffusible IP3 and Ca2+ and replicates Ca2+ wave initiation and spread, as observed in vivo [2]. We tested different distributions and densities of IP3 receptors in the dendrite, assessing the effects on speed and strength of Ca2+ wave boosting. Lowered SERCA pump expression, elevated global IP3R, or reduced spacing between IP3R clusters increased wave spread and speed, altering neuronal excitability. We also assessed propagation efficacy of Ca2+ waves in complex neuronal geometries. Ca2+ wave entry into the soma is impeded due to an impedance-mismatch-like effect. This has implications for Ca2+ dysregulation leading to neurotoxicity: if Ca2+ is more likely to be elevated in dendrites, this would suggest that dendrites may have more susceptibility to apoptotic and ischemic damage.
References
[1] Stutzmann and Mattson. Pharmacol Rev 63(3):700-27, 2011.
[2] Ross WN, Nakamura T, Watanabe S, Larkum M, Lasser-Ross N. Cell and Mol Neurobio 25(2):283, 2005.
The NEURON simulator is a widely used tool for studying multiscale models from the level of ion channels up to the level of large multicolumnar neuronal network models. Although it has been possible to simulate intracellular chemical processes, this has been limited. We have now expanded the tool in order to fully support intracellular chemical dynamics including stochastic reaction-diffusion models and three-dimensional simulations. Arbitrary reaction schemes may be specified at run-time via a Python interpreter, with no separate compilation step required. Models can be imported from Systems Biology Markup Language (SBML) and other resources, facilitating collaboration between the neuroscience and cell biology communities.
A dendritic spine head has a volume of about half a femtoliter, so that only a few particles of a given chemical species, for example calcium ions, are present. By contrast, once these chemicals enter a dendrite, they encounter more uniformity. As calcium ions move randomly around the spine, there is the potential for large percentage deviations from the mean concentration. To study these effects, stochastic simulation is required. We utilize the Gillespie, and tau-leaping algorithms for these simulations, but our methods can easily be extended to other compartment-based approaches. This stochastic simulation must then be coupled to deterministic diffusion to handle the interface with the quasi-continuous environment of the dendrite.
Different chemical reaction-diffusion problems and different geometries require different approximations to solve adequately. For example, previous modelers have implemented radial diffusion as well as longitudinal diffusion, but this missed many biologically important phenomena. Full 3-dimensional spatial simulations has presented a number of challenges. Geometric details are critical; a spatial simulator must define the shape of the joins between dendritic sections. Preliminary modeling shows the value of these techniques to understand how chemical and electrical dynamics interact in neuronal information processing.
AMPAR trafficking plays an essential role in modulating the strength of synaptic transmission. Prior studies have extensively characterized the core signaling underlying AMPAR trafficking, and have identified the cAMP/PKA mediated phosphorylation of GluR1, an AMPAR subunit, as a key step in the membrane insertion of AMPAR. Inhibition of ERK impairs AMPAR membrane insertion, however, the exact mechanism by which ERK exerts its modulation on AMPAR trafficking is still largely unknown. Dopamine (DA), anactivator of PKA and ERK, induces GluR1 membrane insertion. Whereas a fair amount of theoretical work has explored DA signaling leading to AMPAR trafficking, no computational models, to our knowledge, addressed the role of ERK in DA- induced GluR1 trafficking. The present work attempts to fill this gap by proposing a model exploring the regulation of PDE4, a cAMP phosphodiesterase phosphorylated and inhibited by ERK, in DA-mediated GluR1 membrane insertion. We have experimentally validated the DA-induced regulation of PDE4 by ERK and its enhancement of GluR1 phosphorylation and membrane insertion. We find that DA-induced ERK: 1) phosphorylates and inhibits PDE4, increasing cAMP levels, 2) enhances PKA-phosphorylation of DARPP-32, and GluR1 3) decreases PP1 activity and 4) increases GluR1 membrane insertion. These results offer new insight into how ERK modulates PKA-activity during DA-signaling and highlight the complex crosstalk of these two kinase pathways underlying AMPAR trafficking.
The gain of neurons' responses in the auditory cortex is sensitive to contrast changes in the stimulus within a spectrotemporal range very similar to their receptive fields (Rabinowitz 2012), which can be interpreted to represent the tuning of the input to a neuron. This indicates a local mechanism of contrast gain control. Gain control through noisy input has been observed in vitro and in a range of computational models (Ayaz 2009).
We investigate the behaviour of the simplest of such models to showcase gain control, a stochastic leaky integrate-and-fire neuron, which exhibits gain control through divisive normalisation of the input both with and without accompanying subtractive shift of the input-response curve, depending on whether input noise is proportional to or independent of its mean. To get a more direct understanding of how the input statistics change the response, we construct an analytic approximation to the firing rate of our stochastic model neuron, constituted of the expression for the deterministic case, averaged over the derived approximate steady-state distribution of synaptic conductances due to Poissonian synaptic inputs.
This analytic approximation qualitatively produces the same behaviour as simulations and suggests a simple, physiological and local mechanism of contrast gain control in auditory sensing. Building on recent experimental work that has hitherto only been described by phenomenological models (Rabinowitz 2011, 2012), we fit the commonly used sigmoidal response function to simulation data while fixing some parameters with predictions from our model. We obtain good agreement for this lower-dimensional fit, thus providing a structural constraint on the sigmoid parameter choice.
The coding of sensory stimuli in the neural response is a fundamental property of neural systems. Coding determines how many and how accurate a population of neurons can represent stimuli. Although the coding of single stimuli has been studied extensively, coding multiple stimuli at once has been studied far less. Here we study coding in the visual system when stimulus pairs are represented simultaneously. We assume that the neurons code non-linearly to form the response to the pair. Using a maximum-like interaction leads to a higher capacity, than a linear sum. The results provide a novel interpretation for the non-linear interaction observed experimentally. Most coding models assume linear summation of responses [Zemel 1998, Sahani 2003]. Instead maximum interactions have been proposed for object recognition models [Rieserhuber 1999], and circuit models for the maximum function have been proposed [Yu 2002].
