Integrate-and-fire neurons

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Simple neuronal models. Electrical circuit idioms are often used to model neurons. In (a−d ), the entire neuron is reduced to a circuit. The synaptic input is described by as net current I(t). (a) An integrate-and-fire unit. If the voltage V exceeds a fixed threshold, an action potential is fired. (b) Illustration of a leaky integrate-and-fire unit(c) In a rate neuron, discrete pulses are replaced by a continuous output rate. (d) Generally, neurons interact with other neurons directly, so the interaction can be thought of as linear. (e) There are also nonlinear interactions that are mediated by dendritic trees which increase the postsynaptic conductance. Reference: Christof Koch & Idan Segev, Nature Neuroscience 3, 1171 - 1177 (2000). URL:<>

Basic information

The integrate-and-fire neuron is a prime example of describing a neuron based on its membrane potential and has been studied mostly by mathematical methods. In fact, the integrate-and-fire model was one of the first neuronal models to be studied. The French physiologist Louis Lapicque (1866-1952) worked on nerve stimulation in frogs. After looking at the data from his experiment, he introduced a model of the nerve that is based on a simple capacitor circuit. Later scientists built upon his concept and used mathematical methods to analyze the action potential spike that is generated once the neuron is stimulated manually or by synaptic input from presynaptic neurons. The study of these neurons is useful in modeling phenomena in the fields of neurology, physiology, and psychology.

Synaptic Connections

Synaptic Inputs

All synaptic inputs converge onto a single compartment called a point neuron.

Synaptic Outputs

The output consists of a train of asynchronous pulses. In a 'leaky' integrate-and-fire neuron, an ohmic resistance is added to the capacitance, which makes up for the loss of synaptic charge via the resistor and, consequently, the decay of the synaptic input with time.

Spiking properties

These neurons fire action potentials that are separated by time intervals that are of the same orders of magnitude (hundreds of milliseconds).


The neuron works as a circuit consisting of a capacitor C in parallel with a resistor R driven by a current I(t). The driving current can be split into two components, I(t) = IR + IC. The first component is the resistive current IR which passes through the linear resistor R. It can be calculated from Ohm's law as IR = u/R where u is the voltage across the resistor. The second component IC charges the capacitor C.

When these neurons are observed within a network, early studies of the network behavior focused on their suitability as models of associative memory. In these models, the spatial patterns of self-sustained spiking through recurrent connections represent memories that one stores. A number of studies, some of which used a discrete-time formalism, examined the general nature of the patterns of spiking behavior that such networks produce.

Using analytical techniques and computer simulations, certain scientists found that the network behavior is characterized by stable spiking patterns. Furthermore, it was observed that biological neural systems may exhibit coherent oscillations, such as the theta rhythm of the hippocampus and various electroencephalography (EEG) rhythms of the cortex. These might play an important role in neural information processing.


1) Lazar AA, Pnevmatikakis EA. (2008) Faithful Representation of Stimuli with a Population of Integrate-and-Fire Neurons, Neural Computation, 0:0, 1-30.

2) Brunel N, van Rossum MCW. (2007) Lapicque's 1907 paper: from frogs to integrate-and-fire, Biological Cybernetics, 14:49, 1-30.

Additional information

Representation of what occurs in the integrate-and-fire model: Biological neuron model on wikipedia: