### Organizers

It is natural think about the brain and brain function on four different levels: genomics, biochemistry, electrophysiology, and behavior. Enormous amounts of new information are becoming available on associations between genotypes and behavior. The causal mechanisms, which are mostly unknown, necessarily involve the effects of genotype on development, cellular biochemistry, and electrophysiology. The cellular biochemistry and morphology of neurons is fundamental for understanding the electrophysiological properties of neurons and networks. And the network properties then give rise to the brain functions that we label with terms such as memory, mood, decision-making, motor control, and so forth. This simple characterization is misleading because the use of the word "level" suggests that there is bottom up control, the genes control the chemistry that controls the electrophysiology that controls behavior. The scientific issues are so difficult and interesting precisely because this is not true. Behavior affects gene expression levels, electrophysiology induces short and long term changes in cell biochemistry and morphology, which in turn influence the electrophysiology. On each of these levels, mathematicians and computational neuroscientists have created models to give conceptual understanding, to organize data, and to explore causal mechanisms. This workshop will focus on three particular areas. Morphology of neurons and electrophysiological processing. The great variety of dendritic morphologies suggest functional roles for different geometries and it is now understood that dendrites are often not passive conductors. Mathematical models have shown how the distribution of channels and receptor trafficking influence electrophysiological signaling. However, it is also known that electrophysiological signaling affects dendritic processing by affecting synapses and spines and other changes in morphology. For example, gonadotropin-releasing hormone cells of the hypothalamus drive the transition through puberty via changes in cellular- and population-level firing patterns. This change in electrical activity is accompanied by dendritic pruning that alters the electrical/conductance properties of the neuron. Medial superior olive (MSO) neurons in the auditory brainstem decrease their dendritic arborization during postnatal development, eventually achieving bipolar morphology. Mathematical models of the MSO and other neural populations suggest that not only the morphology but also the distribution of different ion channels contributes to dendritic computation. Mathematical models that relate cellular properties to the electrophysiology of neurons often raise new questions in deterministic and stochastic dynamical systems. These include the origins of mixed mode oscillations and bursting behavior, as well as the interplay of stochasticity and synchony. From signaling molecules to behavior. The brain can be in different states with different corresponding behaviors. Signaling molecules play an important role in modulating state, and behavior interacts with the signaling molecules. For instance, the extracellular concentration of the neuromodulator adenosine, which increases during wakefulness and decreases during sleep, appears to increase propensity to transition from waking to sleep by inhibiting wake-active cholinergic cells. In turn, cholinergic cells play a role in inducing REM sleep as well as mediating cognitive functions via signaling on multiple time scales, and acetylcholine has long been recognized as a slow-acting neuromodulator of arousal states. While awake, behavioral activity can provide positive feedback helping to sustain wakefulness in the face of accumulating adenosine. Molecular and electrical signals interact to develop the neuronal network underlying the interactions described above. Moreover, cells have intrinsic mechanisms to tune properties of the electrical signal. For instance, cortical networks have mechanisms that facilitate context-dependent synchronization of different subnetworks within a fixed architecture. In other networks, such as the basal ganglia and spinal interneurons, ion channel mechanisms actively maintain a lack of correlation between nearby cells and this decorrelation may be important for executing smooth movements. Deterministic and stochastic dynamical systems models have been used to investigate the connections between molecular signaling and behavior. Robustness and plasticity. An important property of brain function is that it must continue to operate at each of the four levels outlined in the introductory paragraph despite variation and change in the properties at individual levels. This is true within individuals where properties change as a function of time due to development, meals, emotional and environmental factors, and synapse and cell death. It is also important to understand why system properties are so similar between individuals despite great differences in local detail. And finally, assumptions about the stability of function across species form the basis for conducting experiments on animals and drawing conclusions about human brain function. These kinds of "homeostasis" questions occur in all biological systems, but they are particularly interesting and important in studying brain function for two reasons. First, since neurons are inherently sloppy and stochastic devices, there must be active processes at both the cellular and network level to reliably detect and sharpen electrophysiological signals in the stochastic and noisy environment. Second, one of the most important properties of brain function at all four levels is that it is flexible and changeable on both short and long time-scales. How can brain systems be controlled and homeostatic, yet flexible and changeable at the same time? The answers will be fundamental for understanding brain function and will likely require new advances in stochastic dynamical systems.

### Accepted Speakers

Monday, April 8, 2013 | |
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Time | Session |

08:00 AM | Shuttle to MBI |

08:15 AM 08:45 AM | Breakfast |

08:45 AM 09:00 AM | Welcome, overview of workshop, and introductions: Marty Golubitsky |

09:00 AM 09:45 AM | Antoine Triller - Molecular dynamic at neuronal synapses: from super-resolution to the physical chemistry of molecular interaction 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. |

09:45 AM 10:15 AM | Discussion and Break |

10:15 AM 11:00 AM | Jay Newby - Microtubule transport of mRNA in dendrites 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. |

11:00 AM 11:30 AM | Discussion and Break |

11:30 AM 12:30 PM | Fidel Santamaria - Effects of subcellular dendritic structure on robustness and plasticity of synaptic function 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. |

12:30 PM 02:00 PM | Lunch Break |

02:00 PM 02:45 PM | David Holcman - Synaptic dynamics: modeling, analysis, stochastic simulations and extraction of features from superresolution data of live cell imaging 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. |

02:45 PM 03:15 PM | Discussion and Break |

03:15 PM 04:00 PM | Harel Shouval - Constructing a minimal molecular model of long-term memory 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. |

04:00 PM 04:30 PM | Discussion and Break |

04:30 PM 05:30 PM | Discussion |

05:30 PM 07:00 PM | Reception and poster session in MBI Lounge |

07:15 PM | Shuttle pick-up from MBI |

Tuesday, April 9, 2013 | |
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Time | Session |

08:15 AM | Shuttle to MBI |

08:30 AM 09:00 AM | Breakfast |

09:00 AM 09:45 AM | Mary Kennedy - Modeling calcium influx and biochemical signaling controlled by the NMDA-type glutamate receptor in postsynaptic spines 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. |

09:45 AM 10:15 AM | Discussion and Break |

10:15 AM 11:00 AM | Paul Bressloff - Propagation of CaMKII translocation waves in heterogeneous spiny dendrites 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. |

11:00 AM 11:30 AM | Discussion and Break |

11:30 AM 12:30 PM | William Kath - Synaptic integration in pyramidal neuron dendrites is balanced by distance-dependent synapse distributions and amplification by dendritic spines 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. |

12:30 PM 02:00 PM | Lunch Break |

02:00 PM 02:45 PM | Yulia Timofeeva - Gap junctions, dendrites and resonances: a recipe for tuning network dynamics 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. |

02:45 PM 03:15 PM | Discussion and Break |

03:15 PM 04:00 PM | Steve Cox - Structure-Preserving Model Reduction of Quasi-Active Neurons 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. |

04:00 PM 04:30 PM | Discussion and Break |

04:30 PM 05:30 PM | Discussion |

05:45 PM | Shuttle pick-up from MBI |

Wednesday, April 10, 2013 | |
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Time | Session |

08:15 AM | Shuttle to MBI |

08:30 AM 09:00 AM | Breakfast |

09:00 AM 09:45 AM | Carmen Canavier - Mathematical analysis of depolarization block mediated by slow inactivation of the fast sodium channels that limits phasic signaling in midbrain dopamine neurons. 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. |

09:45 AM 10:15 AM | Discussion and Break |

10:15 AM 11:00 AM | Nicolas Brunel - Calcium-based synaptic plasticity: from single synapses to networks 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. |

11:00 AM 11:30 AM | Discussion and Break |

11:30 AM 12:30 PM | Kyle Miller - The emerging role of forces in axonal elongation 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. |

12:30 PM 02:00 PM | Lunch Break (Pizza provided by MBI) |

02:00 PM 02:45 PM | Richard Bertram - A Hybrid Approach for Understanding Cell Dynamics 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. |

02:45 PM 03:15 PM | Discussion and Break |

03:15 PM 04:00 PM | Gregory Smith - Modeling the bidirectional coupling of localized calcium elevations and whole cell calcium responses 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. |

04:00 PM 04:30 PM | Discussion and Break |

04:30 PM 05:30 PM | Discussion |

05:45 PM | Shuttle pick-up from MBI |

Thursday, April 11, 2013 | |
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Time | Session |

08:15 AM | Shuttle to MBI |

08:30 AM 09:00 AM | Breakfast |

09:00 AM 09:45 AM | Priscilla Greenwood - Genesis of gamma bursts in neural local field potentials 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. |

09:45 AM 10:15 AM | Discussion and Break |

10:15 AM 11:00 AM | Cecilia Diniz Behn - Mathematical modeling of sleep/wake behavior and the role of orexin/hypocretin 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. |

11:00 AM 11:30 AM | Discussion and Break |

11:30 AM 12:30 PM | Bill Lytton - ReactionÂDiffusion Modeling in the NEURON Simulator 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. |

12:30 PM 02:00 PM | Lunch Break |

02:00 PM 02:45 PM | Frances Skinner - Diversity and Details of Hippocampal Interneurons: Contributing to Functional Output 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. |

02:45 PM 03:15 PM | Discussion and Break |

03:15 PM 04:00 PM | Peter Thomas |

04:00 PM 04:30 PM | Discussion and Break |

04:30 PM 05:30 PM | Discussion |

05:45 PM | Shuttle pick-up from MBI |

Friday, April 12, 2013 | |
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Time | Session |

08:15 AM | Shuttle to MBI |

08:30 AM 09:00 AM | Breakfast |

09:00 AM 09:30 AM | Marco Herrera-Valdez - Same cell, different excitability profiles... one dynamical system may not be enough Same cell, different excitability profiles... one dynamical system may not be enough |

09:30 AM 10:00 AM | Emily Stone |

10:00 AM 10:30 AM | Deena Schmidt - Analysis of the stochastic shielding approximation for Markovian ion channel models via random graphs Mathematical models of cellular and sub-cellular physiological mechanisms often involve random walks on graphs representing transitions within networks of functional states. Schmandt and Gal ?an recently introduced a novel stochastic shielding ap- proximation as a fast, accurate method for generating sample paths from a finite state Markov process in which only a subset of states are observable. For example, in ion channel models, such as the Hodgkin-Huxley or other conductance based neural mod- els, a nerve cell has a population of ion channels whose states comprise the nodes of a graph, only some of which allow a transmembrane current to pass. The stochastic shielding approximation consists of neglecting fluctuations in the dynamics associated with edges in the graph not directly affecting the observable states. Here we consider the problem of finding the optimal complexity reducing mapping from a stochastic process on a graph to an approximate process on a smaller sample space, as determined by the choice of a particular linear measurement functional on the graph. The partitioning of ion channel states into conducting versus nonconducting states provides a case in point. In addition to establishing that Schmandt and Gal ?an’s approximation is in fact optimal in a specific sense, we provide heuristic error estimates for the accuracy of the stochastic shielding approximation for an ensemble of Erd ?os- R ?enyi random graphs, using recent results from random matrix theory |

10:30 AM 11:00 AM | Discussion and Break |

11:00 AM 11:30 AM | William Holmes - Things that bug me about single neuron models: Experiences with hippocampal and vestibular neuron modeling 1. Use and abuse of Boltzmann fits 2. Other problems with channel kinetic data 3. Channel time constant expressions do not need to be so complicated and can be intuitive 4. Models that “work” don’t work. Reproducing behavior of vestibular afferents. |

11:30 AM 12:00 PM | Lydia Bilinsky - Slow passage through a Hopf bifurcation in spatially extended excitable systems: some examples from neuroscience Slow passage through a Hopf bifurcation in spatially extended excitable systems: some examples from neuroscience |

12:30 PM | Shuttle pick-up (one to hotel, one to airport) |

Name | Affiliation | |
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Bertram, Richard | bertram@math.fsu.edu | Mathematics Department, Florida State University |

Best, Janet | jbest@math.ohio-state.edu | Mathematics, The Ohio State University |

Bilinsky, Lydia | Mathematics, Duke University | |

Bose, Amitabha | bose@njit.edu | Department of Mathematical Sciences, New Jersey Institute of Technology |

Bressloff, Paul | bressloff@math.utah.edu | Department of Mathematics, University of Utah |

Brunel, Nicolas | nbrunel@galton.uchicago.edu | Statistics and Neurobiology, University of Chicago |

Calvetti, Daniela | daniela.calvetti@case.edu | Department of Mathematics, Case Western Reserve University |

Canavier, Carmen | ccanav@lsuhsc.edu | Cell Biology and Anatomy, Louisiana State University |

Cox, Steve | cox@rice.edu | Computational and Applied Mathematics, Rice University |

Diniz Behn, Cecilia | cdinizbe@gettysburg.edu | Mathematics, Gettysburg College |

Dong, Shuai | dong.209@osu.edu | Pharmacy, The Ohio State University |

Gonzalez, Nancy | nancy.gomath@gmail.com | Instituto de MatemÃ¡ticas, Universidad Nacional Autonoma de Mexico (UNAM) |

Greenwood, Priscilla | pgreenw@math.ubc.ca | Department of Mathematics , University of British Columbia |

Hashemi, Parastoo | phashemi@chem.wayne.edu | Chemistry, Wayne State University |

Herrera-Valdez, Marco | marco.herrera@upr.edu | Applied physics and mathematics, Academia Nacional de InvestigaciÃ³n y Desarrollo |

Holcman, David | holcman@biologie.ens.fr | Department of Physiology, University of San Francisco |

Holmes, William | holmes@ohio.edu | Biological Sciences, Ohio University |

Kath, William | kath@northwestern.edu | Engineering Sciences and Applied Mathematics, Northwestern University |

Kennedy, Mary | kennedym@its.caltech.edu | biology, California Institute of Technology |

Kinzer-Ursem, Tamara | tursem@purdue.edu | Biomedical Engineering, Purdue University |

Lu, Yuanting | ylu@math.fsu.edu | Mathematics, Florida State University |

Lytton, William | Physiology, Pharmacology, Neurology, SUNY Downstate | |

Mangel, Stuart | mangel.1@osu.edu | Neuroscience, The Ohio State University |

Marella, Sashi | sashimarella@gmail.com | Department of Mathematics, New Jersey Institute of Technology |

Matveev, Victor | matveev@adm.njit.edu | Laboratory of Biological Modeling, National Institutes of Health |

Mel, Bartlett | mel@usc.edu | Biomedical Engineering, University of Southern California |

Miller, Kyle | kmiller@msu.edu | Zoology, Michigan State University |

Mitchell, Colleen | colleen-mitchell@uiowa.edu | Mathematics, University of Iowa |

Neves, Susana | susana.neves@mssm.edu | Pharmacology and Systems Thereapeutics, Mount Sinai School of Medicine, CUNY |

Newby, Jay | Mathematical Biosciences Institute, The Ohio State University | |

Raghavachari, Sridhar | raghavachari@neuro.duke.edu | Neurobiology, Duke University Medical Center |

Reed, Michael | reed@math.duke.edu | Mathematics, Duke University |

Rempe, Michael | mrempe@whitworth.edu | Mathematics and Computer Science, Whitworth University |

Santamaria, Fidel | Fidel.Santamaria@utsa.edu | Biology, University of Texas |

Schmidt, Deena | dschmidt@case.edu | Department of Biology & Department of Mathematics, Case Western Reserve University |

Schumacher, Linus | linus.schumacher@maths.ox.ac.uk | Centre for Mathematical Biology, University of Oxford |

Shouval, Harel | harel.shouval@uth.tmc.edu | Neurobiology and Anatomy, University of Texas Health Center |

Skinner, Frances | frances.skinner@utoronto.ca | Toronto Western Research Institute, University Health Network |

Smith, Gregory | greg@as.wm.edu | Applied Science, College of William and Mary |

Stone, Emily | stone@mso.umt.edu | Mathematical Sciences, University of Montana |

Thomas, Peter | peter.j.thomas@case.edu | Neuroscience, Oberlin College |

Timofeeva, Yulia | y.timofeeva@warwick.ac.uk | Computer Science and Centre for Complexity Science, University of Warwick |

Triller, Antoine | triller@biologie.ens.fr | Biologie, Ecole Normale SupÃ©rieure |

Ullah, Ghanim | Theoretical Biology, Los Alamos National Laboratory | |

van Rossum, Mark | mvanross@inf.ed.ac.uk | Informatics, Univ of Edinburgh |

Wallace, Lane | wallace.8@osu.edu | College of Pharmacy, The Ohio State University |

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.

Slow passage through a Hopf bifurcation in spatially extended excitable systems: some examples from neuroscience

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.

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.

Same cell, different excitability profiles... one dynamical system may not be enough

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.

1. Use and abuse of Boltzmann fits

2. Other problems with channel kinetic data

3. Channel time constant expressions do not need to be so complicated and can be intuitive

4. Models that “work” don’t work. Reproducing behavior of vestibular afferents.

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.

Mathematical models of cellular and sub-cellular physiological mechanisms often involve random walks on graphs representing transitions within networks of functional states. Schmandt and Gal ?an recently introduced a novel stochastic shielding ap- proximation as a fast, accurate method for generating sample paths from a finite state Markov process in which only a subset of states are observable. For example, in ion channel models, such as the Hodgkin-Huxley or other conductance based neural mod- els, a nerve cell has a population of ion channels whose states comprise the nodes of a graph, only some of which allow a transmembrane current to pass. The stochastic shielding approximation consists of neglecting fluctuations in the dynamics associated with edges in the graph not directly affecting the observable states.

Here we consider the problem of finding the optimal complexity reducing mapping from a stochastic process on a graph to an approximate process on a smaller sample space, as determined by the choice of a particular linear measurement functional on the graph. The partitioning of ion channel states into conducting versus nonconducting states provides a case in point. In addition to establishing that Schmandt and Gal ?an’s approximation is in fact optimal in a specific sense, we provide heuristic error estimates for the accuracy of the stochastic shielding approximation for an ensemble of Erd ?os- R ?enyi random graphs, using recent results from random matrix theory

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.

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.

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 faithfull

**Structure-Preserving Model Reduction of Quasi-Active Neurons**

Steve Cox

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 mer

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 informatio