The Galves–Löcherbach model (or GL model) is a mathematical model for a network of neurons with intrinsic stochasticity.
In the most general definition, a GL network consists of a countable number of elements (idealized "neurons") that interact by sporadic nearly-instantaneous discrete events ("spikes" or "firings"). At each moment, each neuron "N" fires independently, with a probability that depends on the history of the firings of all neurons since the last time "N" last fired. Thus each neuron "forgets" all previous spikes, including its own, whenever it fires. This property is a defining feature of the GL model.
In specific versions of the GL model, the past network spike history since the last firing of a neuron "N" may be summarized by an internal variable, the "potential" of that neuron, that is a weighted sum of those spikes. The potential may include the spikes of only a finite subset of other neurons, thus modeling arbitrary synapse topologies. In particular, the GL model includes as a special case the general leaky integrate-and-fire neuron model.
Formal definition.
The GL model has been formalized in several different ways. The notations below are borrowed from several of those sources.
The GL network model consists of a countable set of neurons with some set formula_1 of indices. The state is defined only at discrete sampling times, represented by integers, with some fixed time step formula_2. For simplicity, let's assume that these times extend to infinity in both directions, implying that the network has existed since forever.
In the GL model, all neurons are assumed evolve synchronously and atomically between successive sampling times. In particular, within each time step, each neuron may fire at most once. A Boolean variable formula_3 denotes whether the neuron formula_4 fired (formula_5) or not (formula_6) between sampling times formula_7 and formula_8.
Let formula_9 denote the matrix whose rows are the histories of all neuron firings from time formula_10 to time formula_11 inclusive, that is
formula_12
and let formula_13 be defined similarly, but extending infinitely in the past. Let formula_14 be the time before the last firing of neuron formula_15 before time formula_16, that is
 formula_17
Then the general GL model says that
formula_18
Moreover, the firings in the same time step are conditionally independent, given the past network history, with the above probabilities. That is, for each finite subset formula_19 and any configuration formula_20 we have
formula_21
Potential-based variants.
In a common special case of the GL model, the part of the past firing history formula_22 that is relevant to each neuron formula_4 at each sampling time formula_16 is summarized by a real-valued internal state variable or "potential" formula_25 (that corresponds to the membrane potential of a biological neuron), and is basically a weighted sum of the past spike indicators, since the last firing of neuron formula_15. That is,
formula_27
In this formula, formula_28 is a numeric weight, that corresponds to the total weight or strength of the synapses from the axon of neuron formula_29 to the dendrites of neuron formula_15. The term formula_31, the "external input", represents some additional contribution to the potential that may arrive between times formula_10 and formula_33 from other sources besides the firings of other neurons. The factor formula_34 is a "history weight function" that modulates the contributions of firings that happened formula_35 whole steps after the last firing of neuron formula_15 and formula_37 whole steps before the current time.
Then one defines
formula_38
where formula_39 is a monotonically non-decreasing function from formula_40 into the interval formula_41.
If the synaptic weight formula_42 is negative, each firing of neuron formula_29 causes the potential formula_44 to decrease. This is the way inhibitory synapses are approximated in the GL model. The absence of a synapse between those two neurons is modeled by setting formula_45.
Leaky integrate and fire variants.
In an even more specific case of the GL model, the potential formula_44 is defined to be a decaying weighted sum of the firings of other neurons. Namely, when a neuron formula_15 fires, its potential is reset to zero. Until its next firing, a spike from any neuron formula_29 increments formula_44 by the constant amount formula_50. Apart from those contributions, during each time step, the potential decays by a fixed "recharge factor" formula_51 towards zero.
In this variant, the evolution of the potential formula_44 can be expressed by a recurrence formula
formula_53
Or, more compactly,
formula_54
This special case results from taking the history weight factor formula_55 of the general potential-based variant to be formula_56. It is very similar to the leaky integrate and fire model.
Reset potential.
If, between times formula_16 and formula_8, neuron formula_15 fires (that is, formula_60), no other neuron fires (formula_61 for all formula_62),and there is no external input (formula_63), then formula_64 will be formula_65. This self-weight therefore represents the "reset potential" that the neuron assumes just after firing, apart from other contributions. The potential evolution formula therefore can be written also as 
formula_66
where formula_67 is the reset potential. Or, more compactly,
formula_68
Resting potential.
These formulas imply that the potential decays towards zero with time, when there are no external or synaptic inputs and the neuron itself does not fire. Under these conditions, the membrane potential of a biological neuron will tend towards some negative value, the resting or baseline potential formula_69, on the order of −40 to −80 millivolts.
However, this apparent discrepancy exists only because it is customary in neurobiology to measure electric potentials relative to that of the extracellular medium. That discrepancy disappears if one chooses the baseline potential formula_69 of the neuron as the reference for potential measurements. Since the potential formula_44 has no influence outside of the neuron, its zero level can be chosen independently for each neuron.
Variant with refractory period.
Some authors use a slightly different "refractory" variant of the integrate-and-fire GL neuron, which ignores all external and synaptic inputs (except possibly the self-synapse formula_65) during the time step immediately after its own firing. The equation for this variant is
formula_73
or, more compactly,
formula_74
Forgetful variants.
Even more specific sub-variants of the integrate-and-fire GL neuron are obtained by setting the recharge factor formula_51 to zero. In the resulting neuron model, the potential formula_44 (and hence the firing probability) depends only on the inputs in the previous time step; all earlier firings of the network, including of the same neuron, are ignored. That is, the neuron does not have any internal state, and is essentially a (stochastic) function block.
The evolution equations then simplify to 
formula_77
formula_78
for the variant without refractory step, and
formula_79
formula_80
for the variant with refractory step.
In these sub-variants, while the individual neurons do not store any information from one step to the next, the network as a whole still can have persistent memory because of the implicit one-step delay between the synaptic inputs and the resulting firing of the neuron. In other words, the state of a network with formula_81 neurons is a list of formula_81 bits, namely the value of formula_3 for each neuron, which can be assumed to be stored in its axon in the form of a traveling depolarization zone.
History.
The GL model was defined in 2013 by mathematicians Antonio Galves and Eva Löcherbach. Its inspirations included Frank Spitzer's interacting particle system and Jorma Rissanen's notion of stochastic chain with memory of variable length. Another work that influenced this model was Bruno Cessac's study on the leaky integrate-and-fire model, who himself was influenced by Hédi Soula. Galves and Löcherbach referred to the process that Cessac described as "a version in a finite dimension" of their own probabilistic model.
Prior integrate-and-fire models with stochastic characteristics relied on including a noise to simulate stochasticity. The Galves–Löcherbach model distinguishes itself because it is inherently stochastic, incorporating probabilistic measures directly in the calculation of spikes. It is also a model that may be applied relatively easily, from a computational standpoint, with a good ratio between cost and efficiency. It remains a non-Markovian model, since the probability of a given neuronal spike depends on the accumulated activity of the system since the last spike.
Contributions to the model were made, considering the hydrodynamic limit of the interacting neuronal system, the long-range behavior and aspects pertaining to the process in the sense of predicting and classifying behaviors according to a fonction of parameters, and the generalization of the model to the continuous time.
The Galves–Löcherbach model was a cornerstone to the NeuroMat project.
References.
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