Galves–Löcherbach model

3D Vizualization of Galves–Löcherbach model simulating the spiking of 4000 neurons (4 layers with one population of inhibitory neurons and one population of excitatory neurons each) in 180 time intervals.

The Galves–Löcherbach model is a model with intrinsic stochasticity for biological neural nets, in which the probability of a future spike depends on the evolution of the complete system since the last spike.^[1] This model of spiking neurons was developed by mathematicians Antonio Galves and Eva Löcherbach. In the first article on the model, in 2013, they called it a model of a "system with interacting stochastic chains with memory of variable length."

History

Some inspirations of the Galves–Löcherbach model are the 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.^[2] 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.^[3] 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,^[4] the long-range behavior and aspects pertaining to the process in the sense of predicting and classifying behaviors according to a fonction of parameters,^[5][6] and the generalization of the model to the continuous time.^[7]

The Galves–Löcherbach model was a cornerstone to the Research, Innocation and Dissemination Center for Neuromathematics.^[8]

Formal definition

The model considers a countable set of neurons $I$ and models its evolution in discrete-time periods ${\displaystyle t\in \mathbb {Z} }$ with a stochastic chain ${\displaystyle (X_{t})_{t\in \mathbb {Z} }}$, considering values in ${\displaystyle \{0,1\}^{I}}$. More precisely, for each neuron $i\in I$ and time period ${\displaystyle t\in \mathbb {Z} }$, we define ${\displaystyle X_{t}(i)=1}$ if neuron $i$ spikes in period $t$, and conversely ${\displaystyle X_{t}(i)=0}$. The configuration of the set of neurons, in the time period ${\displaystyle t\in \mathbb {Z} }$, is therefore defined as ${\displaystyle X_{t}=(X_{t}(i),i\in I)}$. For each time period ${\displaystyle t\in \mathbb {Z} }$, we define a sigma-algebra ${\displaystyle {\mathcal {F}}_{t}=\sigma (X_{s}(j),j\in I,s\leq t)}$, representing the history of the evolution of the activity of this set of neurons until the relevant time period $t$. The dynamics of the activity of this set of neurons is defined as follows. Once the history ${\displaystyle {\mathcal {F}}_{t-1}}$ is given, neurons spike or not in the next time period ${\displaystyle t}$ independently from one another, that is, for each finite subset ${\displaystyle F\subset I}$ and any configuration ${\displaystyle a_{i}\in \{0,1\},i\in F,}$ we have

${\displaystyle {\mathop {\mathrm {Prob} }}(X_{t}(i)=a_{i},i\in I|{\mathcal {F}}_{t-1})=\prod _{i\in I}{\mathop {\mathrm {Prob} }}(X_{t}(i)=a_{i}|{\mathcal {F}}_{t-1}).}$

Furthermore, the probability that a given neuron $i$ spikes in a time period $t$, according to the probabilistic model, is described by the formula

${\displaystyle {\mathop {\mathrm {Prob} }}(X_{t}(i)=1|{\mathcal {F}}_{t-1})=\phi _{i}{\Biggl (}\sum _{j\in I}W_{j\rightarrow i}\sum _{s=L_{t}^{i}}^{t-1}g_{j}(t-s)X_{s}(j),t-L_{t}^{i}{\Biggl )},}$

where ${\displaystyle W_{j\rightarrow i}}$ is synaptic weight that expresses the increase of membrane potential of neuron $i$ because of neuron $j$'s spike, $g_{j}$ is a function that models the leak of potential and ${\displaystyle L_{t}^{i}}$ is the most recent period of neuron $i$'s spike before the given time period $t$, considering the formula

${\displaystyle L_{t}^{i}={\text{sup}}\{s<t:X_{s}(i)=1\}.}$

At time $s$ before $t$, neuron $i$ spikes, restoring the membrane potential to its initial value.

See also

• Biological neuron model
• Hodgkin-Huxley model
• Computational neuroscience
• NeuroMat

References

1. A. Galves, E. Löcherbach, "Infinite Systems of Interacting Chains with Memory of Variable Length — A Stochastic Model for Biological Neural Nets". Journal of Statistical Physics, vol. 151, n. 5, pp. 896–921, Jun. 2013
2. B. Cessac, "A discrete time neural network model with spiking neurons: II: Dynamics with noise". Journal of Mathematical Biology, Vol. 62, nº 6, pg 863–900. Jun. 2011
3. H. E. Plesser, W. Gerstner. "Noise in Integrate-and-Fire Neurons: From Stochastic Input to Escape Rates". Neural Computation. Feb 2000, Vol. 12, No. 2, Pg 367–384
4. A. De Masi, A. Galves, E. Löcherbach, E. Presutti, "Hydrodynamic limit for interacting neurons". Journal of Statistical Physics, 158(4), 866–902, 2015.
5. A. Duarte, G. Ost, "A model for neural activity in the absence of external stimuli", arXiv preprint arXiv:1410.6086 (2014).
6. N. Fournier, E. Löcherbach, "On a toy model of interacting neurons", arXiv preprint arXiv:1410.3263 (2014).
7. K. Yaginuma, "A stochastic system with infinite interacting components to model the time evolution of the membrane potentials of a population of neurons", arXiv preprint arXiv:1505.00045 (2015).
8. "Modelos matemáticos do cérebro", Fernanda Teixeira Ribeiro, Mente e Cérebro, Jun. 2014