Workshop 5: Cellular and Subcellular

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  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 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.

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  Mathematical analysis of depolarization block mediated by slow inactivation of the fast sodium channels that limits phasic signaling in midbrain dopamine neurons.
  Carmen Canavier

  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.

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  Modeling calcium influx and biochemical signaling controlled by the NMDA-type glutamate receptor in postsynaptic spines
  Mary Kennedy

  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.