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Free C++ source code for the integrate and fire neuron model is available.
How Neurons are Modeled
Click here for a detailed list neuron properties.
The neuron is modeled as a RC (resistor-capacitor) circuit, in which the user
sets the time constant. The nominal input resistance is set at 1 megohm, but
this can be altered by changing the relative electrical size. Doubling the size
effectively halves the input resistance. The neuron is perturbed from its
resting membrane potential by current passing across the membrane. There are 3
sources of current.
| 1. Current can be injected by the user as if through a
microelectrode inserted into the neuron. |
| 2. Current can pass into the neuron
from another neuron to which it is coupled by an electrical synapse. The effect
of current from either of these sources depends upon the electrical size of the
neuron; the larger the size the smaller the voltage perturbation. This follows
from Ohm's law, since the current itself does not change with neuron size. |
| 3.
Current can pass through a chemical synaptic conductance. The voltage
perturbation produced by this current is independent of size, since in the
simulation the synaptic current scales proportionately with size (i.e. ion
channel density is assumed to remain constant as the neuron changes in size). |
Spiking Properties
Each integrate-and-fire neuron has a spike threshold. If the membrane potential exceeds this
threshold (i.e. is more positive), the neuron spikes. A spike is modeled as a
brief (1 integration time step) shift in membrane potential to the defined spike
peak amplitude. NOTE: no current flow is modeled during the depolarising phase
of the spike, and thus the spike is largely "cosmetic". This is to increase the
speed of the simulation, but it can reduce the realism of effects mediated by
electrical and non-spiking chemical synapses. The main problem is that the spike
is shorter in duration than it may be in reality, which means that less current
flows in an electrical synapse. To help overcome this limitation a “strength”
parameter is defined, which simply changes the effective spike
amplitude, without changing the displayed amplitude. A spike in a neuron is
followed by a sudden increase in membrane conductance with amplitude defined by
the Afterspike Hyperpolarising Potential (AHP) conductance parameter. This
conductance increase is to an ion with an equilibrium potential defined by the
AHP equilibrium potential parameter, and would normally be set to a value
appropriate for potassium ions. The AHP conductance then decays exponentially
back to a zero value, with a defined AHP time constant. Following a spike there
is also a brief defined absolute refractory period, during which the threshold
is set infinitely high.
A relative accommodation level can be defined for spike threshold. Accommodation
means that spike threshold potential varies as membrane potential itself varies.
An accommodation level of 0 means no accommodation; the threshold is fixed at
its defined initial value. An accommodation level of 1 means that when the
membrane potential changes, the threshold also changes so as to eventually
exactly parallel the membrane potential, and maintain the same relative voltage
difference as is defined between the resting potential and the initial
threshold. However, the spike threshold does not change instantaneously, but
rather approaches its new value with an exponential time course, as defined by
the accommodation time constant. Thus rapid changes in membrane potential can
induce spiking, even if the relative accommodation level is large, so long as
the accommodation time constant is long.
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1. Initial threshold level. |
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2. Accommodated threshold level. |
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3. Accommodation time constant. |
A simple voltage and time-dependent inactivating calcium current is implemented
which, if the appropriate parameters are defined, can give a neuron endogenous
bursting or plateau capabilities. The calcium current has a defined equilibrium
potential and maximum conductance. Activation and inactivation both have
first-order kinetics (i.e. mh, as opposed to m3h for the Hodgkin-Huxley model).
Each variable has a defined mid-point voltage at which it has a steady-state
value of 0.5, a mid-point slope which describes the rate of change of the
variable with voltage at the mid-point, and a time constant, which describes the
rate at which the variable approaches its steady-state value.
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