Figure Code examples.all

The Dual Constraint model applied to the mouse lumbrical muscle. Left hand column shows the development through time of the pattern of connectivity as displayed in a connection matrix C_nm. The area of each square indicates the strength of that element of C_nm.  For clarity, only twenty of the 240 endplates are shown. Right hand column shows the development through time of four different quantities. (a) The value of C_n1, i.e. the strength of connections from each of the six axons competing for the first muscle fibre (Equation 10.13). (b) The value of the amount of available presynaptic resource A_n at each of the six axons (Figure 10.14). (c) The motor unit size ν_n of each of the six axons. (d) The number of endplates μ which are singly, or multiply innervated. The colour of the line indicates the level of multiple innervation, ranging from 1 (black) to 6 (blue). Parameters: N = 6, M = 240, A₀ = 80, B₀ = 1,  α = 45, β = 0.4, γ = 3, δ = 2. Only terminals larger than θ = 0.01 were considered as being viable. Terminals that were smaller than this value were regarded as not contributing to νn and therefore did not receive a supply of  A.

The time courses of membrane potential,  conductances and gating variables during an action potential.

Voltage response to a single synaptic EPSP at a spine on a dendrite. There are three spines 100 μm apart on a 1000 μm long dendrite of 2 μm diameter. EPSP occurs at spine 1. Black lines show the voltage at each spine head. Blue lines are the voltage response in the dendrite at the base of each spine. Dendrite modelled with 1006 compartments (1000 for main dendrite plus 2 for each spine); passive membrane throughout with R_m = 28 kΩ cm², Ra = 150 Ω cm, C_m = 1 μF cm⁻²; spine head: diameter 0.5 μm, length 0.5 μm; spine neck: diameter 0.2 μm, length 0.5 μm. For a similar circuit see Woolf et al. (1991).

The effect of the A-type current on neuronal firing in simulations. (a) The time course of the membrane potential in the model described in Box 5.2 in response to current injection of 8.21 µA cm² that is just superthreshold. The delay from the start of the current injection to the neuron firing is over 300 ms. (b) The response of the model to a just suprathreshold input (7.83 µA cm²) when the A-type current is removed. The spiking rate is much faster. In order to maintain a resting potential similar to the neuron with the A-type current, the leak equilibrium potential E_L is set to -72.8 mV in this simulation. (c) A plot of firing rate f versus input current I in the model with the A-type conductance shown in (a).  Note the gradual increase of the firing rate just above the threshold, the defining characteristic of Type I neurons. (d) The f-I plot of the neuron with no A-type conductance shown in (b). Note the abrupt increase in firing rate, the defining characteristic of Type II neurons.

Tail currents in the Vandenberg and Bezanilla (1991) model (VB, black traces) and the Hodgkin and Huxley (1952d) model (HH, blue traces) of the sodium current in squid giant axon in response to the voltage-clamp protocol shown at the base, where the final testing potential could be (a) -98mV or (b) -58mV. During the testing potential, the tail currents show a noticeably slower rate of decay in the VB model than in the HH model. To assist comparison, the conductance in the VB model was scaled so as to make the current just before the tail current the same as in the HH model. Also, a quasi-ohmic I-V characteristic with the same E_Na as the HH model was used rather than the GHK I-V characteristic used in the VB model. This does not affect the time constants of the currents, since the voltage during each step is constant.

Simulations of patches of membrane of various sizes with stochastic Na⁺ and K⁺ channels. The density of Na⁺ and K⁺ channels is 60 channels per µm² and 18 channels per µm² respectively, and the single channel conductance of both types of channel γ = 2 pS. The Markov kinetic scheme versions of the Hodgkin-Huxley potassium and sodium channel models were used (Schemes 5.5 and 5.8). The leak conductance and capacitance is modelled as in the standard HH model. No current is applied; the action potentials are due to the random opening of a number of channels taking the membrane potential to  above threshold.

Calcium transients in a 1.0 single cylindrical compartment, 1 μm in diameter and 1 μm in length (similar in size to a spine head). Initial concentration is 0.05 μM. Influx is due to a calcium current of 5 μA cm⁻², starting at 20 ms for 2 ms, across the radial surface. Black line: accumulation of calcium with no decay or extrusion. Blue line: simple decay with τ_dec = 27 ms. Gray dashed line: instantaneous pump with V_pump = 4 × 10⁻⁶ μMs⁻¹, equivalent to 10⁻¹¹ mol cm⁻² s⁻¹ through the surface area of this compartment, K_pump = 10 μM, J_leak = 0.0199 × 10⁻⁶ μM^s−1.

Calcium transients in (a) the potassium channel colocalised pool, (b) the larger submembrane shell, (c) the compartment core in a three pool model. Note that in (a) the concentration is plotted on a different scale. Compartment 1 μm in diameter with 0.1 μm thick submembrane shell; colocalised membrane occupies 0.001% of the total membrane surface area. Calcium influx into the colocalised pool. Calcium current, diffusion and membrane-bound pump parameters as in Figure 6.5.

Longitudinal diffusion along a 10 μm length of dendrite. Plots show calcium transients in the 0.1 μm thick submembrane shell at different positions along the dendrite. Solid black line: 0.5 μm; dashed: 1.5 μm; dot-dashed: 4.5 μm. (a) Diameter is 1 μm. (b) Diameter is 4 μm. Calcium influx occurs in the first 1 μm length of dendrite only. Each of ten 1 μm long compartment contains four radial shells. Blue dotted line: calcium transient with only radial (and not longitudinal) diffusion in the first 1 μm long compartment. All other model parameters as in Figure 6.5.