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Dendritic Modelling 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. 80% of all excitatory synapses - at the dendritic spines. 80% of all excitatory synapses - at the dendritic spines. learning and memory, logical computations, pattern matching, amplification of distal synaptic inputs, temporal filtering. 80% of all excitatory synapses - at the dendritic spines. learning and memory, logical computations, pattern matching, amplification of distal synaptic inputs, temporal filtering. 80% of all excitatory synapses - at the dendritic spines. learning and memory, logical computations, pattern matching, amplification of distal synaptic inputs, temporal filtering. 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. 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 ( τ ) and space constant ( λ ) simplify the temporal and spatial description of signal decay. dendrite segments of different diameters, lengths, and electrical properties. However, it is impor- tant 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. Dendritic modelling2 The objective of realistic single cell modelling 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 framework of cable theory and compartmental modelling. The theoretical basis of compartmental modelling 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 (τ) and space constant (λ) simplify the temporal and spatial description of signal decay. 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 ∂2V = −V + ∂T ∂X2 where V = Vm−Vr represents the departure of the membrane potential V (which is the difference, intracellular Vi minus extracellular Ve), from its resting value, Vr. Also, x X = , λ = r /r = (R /R )(d/4), λ m i m i t ! ! T = , τm = rmcm = RmCm. τm Here ri is the intracellular (core) resistance per unit length of the cylinder, while cm and 1/rm are the membrane capacity, and membrane conductance, respectively, per unit length of the membrane cylinder; d is the cylinder diameter, while Rm and Cm apply to unit membrane area, and Ri 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. 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. 2from the article by Dieter Jaeger: http://www.brains-minds-media.org/archive/222/ 3 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. Signal decay can be depicted graphically by “morphoelectronic transforms”. Here it is demon- strated 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. Another important feature of signal decay is its dependence on input resistance, which itself decreases dramatically during barrages with synaptic input. Thus the shown morphoelectronic transforms would show much more pronounced signal decay yet if the neuron was subjected to a synaptic input barrage. Beyond the passive structure of a neuron (process diameters, lengths, passive input resistance, membrane capacitance, and axial resistance) neural dynamics are crucially dependent on the presence of voltage-gated ion channels. The modelling of individual ion channels would be too computationally costly, however, and the Hodgkin-Huxley formalism of macroscopic currents (one for each species of ion channel) is typically employed. The overall structure of a compartmental model is depicted. The neuron is divided into isopoten- tial compartments (these need to be chosen small enough such that signal decay within the length of one compartment is expected to be negligible under all conditions), and each compartment is given passive properties: membrane capacitance (Cm), membrane resistance (described by leak 4 dendrite segments of different diameters, lengths, and electrical properties. However, it is impor- tant 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. Dendritic modelling2 The objective of realistic single cell modelling 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 framework of cable theory and compartmental modelling. The theoretical basis of compartmental modelling 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 (τ) and space constant (λ) simplify the temporal and spatial description of signal decay. Because
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