80% of all excitatory synapses - at the dendritic spines.

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1 Dendritic Modelling Dendrites (from Greek dendron, tree ) are the branched projections of a neuron that act to conduct the electrical stimulation received from other cells to and from the cell body, or soma, of the neuron from which the dendrites project. Electrical stimulation is transmitted onto dendrites by upstream neurons via synapses which are located at various points throughout the dendritic arbor. Dendrites play a critical role in integrating these synaptic inputs and in determining the extent to which action potentials are produced by the neuron.

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3 80% of all excitatory synapses - at the dendritic spines.

4 80% of all excitatory synapses - at the dendritic spines. learning and memory, logical computations, pattern matching, amplification of distal synaptic inputs, temporal filtering.

5 80% of all excitatory synapses - at the dendritic spines. learning and memory, logical computations, pattern matching, amplification of distal synaptic inputs, temporal filtering.

6 80% of all excitatory synapses - at the dendritic spines. learning and memory, logical computations, pattern matching, amplification of distal synaptic inputs, temporal filtering.

7 Electrical properties of dendrites The structure and branching of a neuron's dendrites, as well as the availability and variation in voltage-gated ion conductances, strongly influences how it integrates the input from other neurons, particularly those that input only weakly. This integration is both temporal -- involving the summation of stimuli that arrive in rapid succession -- as well as spatial -- entailing the aggregation of excitatory and inhibitory inputs from separate branches. Dendrites were once believed to merely convey stimulation passively, i.e. via diffusive spread of voltage and without the aid of voltage-gated ion channels. Passive cable theory describes how voltage changes at a particular location on a dendrite transmit this electrical signal through a system of converging dendrite segments of different diameters, lengths, and electrical properties. However, it is important to remember that dendrite membranes are host to a zoo of proteins some of which may help amplify or attenuate synaptic input (like those in the axon). One important feature of dendrites, endowed by their active voltage gated conductances, is their ability to send action potentials back into the dendritic arbor. Known as backpropagating action potentials, these signals depolarize the dendritic arbor and provide a crucial component toward synapse modulation and long-term potentiation.

8 The dendrite as a stretched-out RC circuit The objective of realistic single cell modeling is to approximate the electrophysiological behavior of biological neurons as closely as possible. Since each type of neuron in the brain is characterized by distinct morphological features, as well as distinct electrophysiological features this requires a case-by-case study -- within the overall famework of cable theory and compartmental modelling. The theoretical basis of compartmental modeling is given by cable theory, a set of well understood concepts originally introduced by Wilfried Rall. A basic dictum of cable theory is that in passive cylinders signals decay over space and time in a mathematically understood fashion. The concepts of time constant ( constant ( λ ) simplify the temporal and spatial description of signal decay. τ ) and space

9 Because passive membrane is assumed, the spatio-temporal distribution of membrane potential along the cylinder must obey a partial differential equation, known as the cable equation; this can be expressed V T = V + 2 V X 2 where V = V m V r represents the departure of the membrane potential V (which is the difference, intracellular V i minus extracellular V e ), from its resting value, V r. Also, X = x λ, λ = r m /r i = (R m /R i )(d/4), T = t τ m, τ m = r m c m = R m C m. Here r i is the intracellular (core) resistance per unit length of the cylinder, while c m and 1/r m are the membrane capacity, and membrane conductance, respectively, per unit length of the membrane cylinder; d is the cylinder diameter, while R m and C m apply to unit membrane area, and R i is the volume resistivity of the intracellular medium. In practice, neurons have finite length processes with closed ends. Steady state voltage decays significantly when processes are of length > 1 space constant or so, which is typically the case in large neurons.

10 Membrane decay constant τ. Comparison of normalized charging curves for finite cables of different electronic lengths. A step of current is injected and the voltage is measured, at x = 0. e x/λ Membrane space constant λ. Comparison of voltage decays along finite cables of different electronic lengths and with end terminations. Current is injected at x = 0. The solid lines are for finite cables with sealed ends (no current flows out of either end), the dashed lines are for finite cables with open ends, and the dotted line is for a semi-infinite cable.

11 Morphoelectric transforms Importantly, signal decay in cylinders is frequency dependent, and signals with fast rise and fall time (such as action potentials) decay over much shorter distances than steady state potentials. Signal decay can be depicted graphically by morphoelectronic transforms. Here it is demonstrated that signal decay is non-symmetrical and that voltage transients generated in the dendrites decay much more than voltage transients from the soma to the tip of the dendrites. This effect is due to the branching structure of the neuron and the increasing diameter of processes toward the soma.

12 Compartmental modelling Beyond the passive structure of a neuron neural dynamics are crucially dependent on the presence of voltage-gated ion channels. The modeling of individual ion channels would be too computationally costly, however, and the Hodgkin-Huxley formalism of macroscopic currents is employed. In a compartmental model the neuron is divided into isopotential compartments, and each compartment is given both passive and active properties. The three steps to make a single neuron model are i) create an accurate morphological reconstruction, ii) create an accurate passive model, and iii) match active properties with physiological data.

13 i) Create an accurate morphological reconstruction Biocytin fills from slice recordings.

14 Creating a model starts with a morphological reconstruction of dye-filled neurons. Typically biocytin fills from slice recordings are used. The initial histological processing can already lead to distortions of the true neural morphology due to fixation shrinkage as well as mechanical stresses. Here, the same Purkinje cell is shown before and after cover-slipping. A little bit of squeeze on the cover slip significantly distorts the relative branch lengths. A certain amount of error in morphological reconstructions due to such distortions as well as incomplete fills of spines and distal processes should always be expected. Alternatively grab data from an existing database such as Hippocampal Neuronal Morphology Archive at Cell Centered Database (CCDB) at Virtual Neuromorphology electronic database at Neocortical microcircuit cell database at Also it is worth knowing about Cvapp this is a cell viewing, editing and format converting program for morphology files. It can also be used to prepare structures digitized with Neurolucida software for modelling with NEURON. It is available from

15 ii) Create an accurate passive model After the morphology description has been imported into the simulation software, the passive parameters of the cell need to be fit to data. If the morphological reconstruction is less than ideal, the fit parameters may show a shift with respect to the true values to offset erroneous measurements on cell surface area and process diameters. For example, if surface area is missing, the specific membrane capacitance may come out higher than the standard value of 1µF cm 2. Tasks: Set R m, C m, and R i correctly for each compartment. If the assumption that parameters are uniform holds, need to fit 3 values. Strategy: Obtain data from recordings, then optimize R m, C m, and R i to fit data. To fit passive parameters, recorded neural responses of the same cell type two current injection pulses need to be available. These recordings need to be undertaken after voltage-gated conductances are blocked. A complete block is often difficult, and the recorded passive responses may deviate from the true passive response due to several other potential problems as well (such as due to the electrode serial resistance). Fitting passive responses is done by running current injections into the soma of the model cell and optimizing the match of the voltage response with that of recordings.

16 iii) Match active properties with physiological data This takes the bulk of the total time in making a good model. The construction of a biologically realistic model is a more hopeful prospect if the availability of voltage-clamp date for the various voltage-gated conductances of the cell type under study is good. If such data are not available, one can proceed with default channel kinetics determined in other cell types and fit current-clamp data as well as possible. However, this latter process is unlikely to generate an accurate model in the end. Tasks: Construct voltage-gated conductances with correct kinetics. Incorporate intracellular calcium handling. Find needed densities of channels in each compartment. Add realistic synaptic input conductances. Once a model is complete, it can help the scientist to appreciate the 3-dimensional distribution of voltage and currents throughout different activity regimes. Movie files are one helpful tool to obtain this information.

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