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Sandpile model

Sandpile model

Here we describe the sandpile model which was proposed by Bak in collaboration with Tang and Wiesenfeld [1,2]. It attracted great interest as the first and clearest example of self-organized criticality.

The model is inspired by a real pile of sand. Grains of the model ``sand'' are dropped into the system and are lost at the boundaries, allowing the system to reach a stationary state that balances input and output. In the limit of infinitely slow input, the system displays a highly fluctuating, scale-invariant avalanche-like pattern of activity.

Formally, the sandpile model is defined on the $ d$-dimensional lattice of size $ L^d$. The variable $ z(\mathrm{r})$ describes the ``energy'' of the site $ \mathrm{r}\in \mathbb{R}^d$, and $ e_j$, $ j=1,\ldots,d$ are basis vectors on the lattice. The dynamics of $ z$ obeys the following rules:
1) If for all sites of the latice $ \mathrm{r}$ , $ z(\mathrm{r_k})<z_c$, then one site $ \mathrm{r}_i$ is selected randomly and one grain of sand is dropped there

$\displaystyle z(\mathrm{r}_i) \to z(\mathrm{r}_i)+1.$ (0.1)

2) If $ z(\mathrm{r}_i) \geq z_c$, then site $ \mathrm{r}_i$ relaxes by the following rule:

$\displaystyle z(\mathrm{r}_i)$ $\displaystyle \to$ $\displaystyle z(\mathrm{r}_i)-2d$  
$\displaystyle z(\mathrm{r}_i\pm e_j)$ $\displaystyle \to$ $\displaystyle z(\mathrm{r}_i \pm e_j)+1 \:\:\:\: \mathrm{ for } \:\:j=1,\ldots,d.$  

Image sanpilefall

On the boundary, energy dissipates from the system and there is no other form of dissipation in the system. At each moment of time, only one of the rules is used. This creates an infinite separation of timescales of the internal relaxation (rule 2) and the external input (rule 1).

An avalanche is then a sequence of relaxations that directly follows the addition of one grain. The system reaches a stationary state that is characterized by a power-law distribution of avalanches sizes [1,14,22]

$\displaystyle P(s)\sim s^{-\gamma}.$    

Earthquakes

When people started measuring the magnitudes of earthquakes, they found that there were a lot more small earthquakes than large ones. On the right, you can see a plot of all the earthquakes of magnitude 4 or greater in 1995. You can see that there are a few really big earthquakes, and many many small earthquakes.
Image 1995_earthquakes

Image 1995_earthquakes_histogram
Geologists have found that the number of earthquakes of magnitude M is proportional to 10-bM. They call this law the Gutenberg-Richter law [12]. Look at the graph of all the earthquakes in 1995 on the left: the red line gives the Gutenberg-Richter prediction with b = 1. The value of b seems to vary from area to area, but worldwide it seems to be around b=1.

Power-law statistics of earthquakes size suggests that the tectonic plates are in self-organized critical state.

Neuronal avalanches

First experiments were done in cultured and acute cortical slices by John Beggs and Dietmar Plenz [4,5,25,26]. Cultures were planted on a multielectrode array and local field potential signals were recorded from the 64 electrodes of the array during a long period of time (on a time scale of hours). A local field potential (LFP) is a signal which reflects the sum of all synaptic activity within a volume of tissue.

Figure: Extraction of LFPs from a multielectrode array. 
\includegraphics[width=0.95\columnwidth]{Pictures/Figure1.eps}

In Fig. 1, the extraction of the filtered LFP signal from the cortical slice is shown. The first filtering stage extracts the LFPs from the recorded signal, which are then thresholded to obtain the binary signal on each electrode. The data is then organized in 4 ms bins. After such processing, the data consists of short intervals of activity, when one or more electrodes detected LFPs above the threshold, separated by longer periods of silence. When the activity periods are studied in fine temporal resolution, it is possible to see the spread of activity in the slice. A set of consecutively active frames is called an avalanche. The size of an avalanche is defined as the number of electrodes which were active during the avalanche. Avalanche sizes turn out to follow the power-law distribution with the exponential cutoff at the size of the multielectrode array. The distribution is stable over many hours. The exponent of $ -3/2$ characterizes both acute cortical slices and cultures.

The application of the inhibition antagonist picrotoxin makes the network more exitable and changes the distribution of the avalanche sizes to bimodal. After a few days, the network returns to the critical state despite pharmacological influence (Fig. 2). This implies that there are some long-term regulatory mechanisms which tune the network to criticality.

In the cortex, the emergence of power-law distribution of avalanche sizes with slope $ -3/2$ depends on an optimal concentration of dopamine [26] and on the balance of excitation and inhibition [4,26], which suggests that particular parameters must be appropriately ``tuned'' [3]. This provides additional evidences in favor of the hypothesis that there are mechanisms in the cortex which lead to robust and local self-organization towards the critical state.

Despite the complex relationship between LFP waveform and underlying neuronal processes, neuronal avalanches have also been recently identified in vivo in the normalized LFPs extracted from ongoing activity in awake macaque monkeys [24]. This proves that critical avalanches are not an artifact of the neuronal cultures and slices and also present in the brain of the behaving animals.

Until recently, avalanches were observed only in LFP recordings, however two publications appeared last year, in which critical avalanches were measured in spikes. In the dissociated cultures of rat hippocampal neurons and intact leech ganglia, avalanche sizes have a power-law distribution with slope $ -3/2$ [23]. In cortical culture with an array of 500 electrodes, a power law with an exponent of approximately $ -2.1$ was observed [3].

Figure 2: Avalanche sizes and durations distribution. (left) Black: avalanche size distribution in unperturbed system, red: avalanche size distribution after application of picrotoxin, gray: relaxation of distribution after washing out of the reagent, dashed line: power-law with exponent -3/2. (right) Distribution of avalanche durations, dashed line: power-law with exponent -2. Pictures are taken from [4]
\includegraphics[width=0.5\columnwidth]{Pictures/beggs2.eps}                   \includegraphics[width=0.5\columnwidth]{Pictures/beggs_duration.eps}

Question of adequate modelling of SOC in neural networks is very recent. There are solutions based on the short-term synaptic plasticity [19,20,18,8], on the long-term synaptic plasticity [17] and on the specific network structure [27]. But there is still large space for further models that will better describe and predict observed phenomena.


Anna Levina and J. Michael Herrmann