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Journal of Computational Neuroscience

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01-10-2011

Leader neurons in leaky integrate and fire neural network simulations

Author: Cyrille Zbinden

Published in: Journal of Computational Neuroscience | Issue 2/2011

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Abstract

In this paper, we highlight the topological properties of leader neurons whose existence is an experimental fact. Several experimental studies show the existence of leader neurons in population bursts of activity in 2D living neural networks (Eytan and Marom, J Neurosci 26(33):8465–8476, 2006; Eckmann et al., New J Phys 10(015011), 2008). A leader neuron is defined as a neuron which fires at the beginning of a burst (respectively network spike) more often than we expect by chance considering its mean firing rate. This means that leader neurons have some burst triggering power beyond a chance-level statistical effect. In this study, we characterize these leader neuron properties. This naturally leads us to simulate neural 2D networks. To build our simulations, we choose the leaky integrate and fire (lIF) neuron model (Gerstner and Kistler 2002; Cessac, J Math Biol 56(3):311–345, 2008), which allows fast simulations (Izhikevich, IEEE Trans Neural Netw 15(5):1063–1070, 2004; Gerstner and Naud, Science 326:379–380, 2009). The dynamics of our lIF model has got stable leader neurons in the burst population that we simulate. These leader neurons are excitatory neurons and have a low membrane potential firing threshold. Except for these two first properties, the conditions required for a neuron to be a leader neuron are difficult to identify and seem to depend on several parameters involved in the simulations themselves. However, a detailed linear analysis shows a trend of the properties required for a neuron to be a leader neuron. Our main finding is: A leader neuron sends signals to many excitatory neurons as well as to few inhibitory neurons and a leader neuron receives only signals from few other excitatory neurons. Our linear analysis exhibits five essential properties of leader neurons each with different relative importance. This means that considering a given neural network with a fixed mean number of connections per neuron, our analysis gives us a way of predicting which neuron is a good leader neuron and which is not. Our prediction formula correctly assesses leadership for at least ninety percent of neurons.

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Negative axon lengths are not allowed and we impose a maximum length of \(\frac{2}{3}\sqrt{N}\) for each axon. With this last condition, no axon can cover all the network in any direction. The rationale for this choice is that, in the experiments (Eckmann et al. 2008), probably no axon covers all the test tube in any given direction.

Random network in which the probability of connections between all pairs of neurons is equal are also studied in this paper. Table 3 shows also results concerning this particular kind of random networks.

We know that the membrane potential fluctuates more or less between − 70 and − 50 mV (Gerstner and Kistler 2002). But to simplify, we will approximately scale the membrane potential between 0 and 1.

We decided that negative membrane potential firing thresholds are not allowed since we do not want the firing condition \(\left(V_n(t)>V^{*}_{n}\right)\) to be reached for some neurons each time they are reset.

Here, by event we mean the moment when a neuron receives a signal from another neuron or when a neuron receives some noise (i.e., the Markovian exponential clock Ω_t,n rings).

The mean membrane potential charge due to the noise is \(\sigma+\sigma e^{-\frac{\omega}{\tau}}+\sigma e^{-2\frac{\omega}{\tau}}+\ldots=\frac{\sigma}{1-e^{-\frac{\omega}{\tau}}}\cong\frac{\tau\sigma}{\omega}\) if \(\frac{\omega}{\tau}\) is small.

Unbiased estimator of the considered probability.

It would be exactly the binomial distribution if all the spikes were independent. All the spikes are certainly not independent because neurons are connected. So the binomial distribution is not the correct one but gives a good enough approximation (see Zbinden 2010).

Here by “reasonable” parameters, we mean to observe a neural activity more or less like the one in Fig. 2 but without a precise predicting spike timing (Brette and Gerstner 2005; Luscher et al. 2006). This means that we want to observe quiet periods, successful pre-bursts and bursts in a way to find leaders. Note that this range of “reasonable” parameters is quite large and the relation between the dynamical parameters is non trivial, see remark in Sections 2.2 and 4.1.

This means a random network in which the probability of connections between all possible pairs of neurons is equal.

Remark that in the special case where ω ≪ τ, we rarely observed a few inhibitory neurons with a leadership score higher than 3. This can happen only when these inhibitory neurons have a very low membrane potential firing threshold, a few excitatory fathers and a lot of inhibitory fathers. The reason is as follow, because ω ≪ τ, the network gets synchronized after the first burst and these particular inhibitors have the faculty to fire even before the true trigger of the burst. Beside the type of their fathers implies a very low neural activity, so their leadership scores are potentially good. Note that the few best leaders are always excitatory neurons.

The neural activity ratio of neuron n, noted q _n, is defined in Section 3.2.2.

This means that we choose N, r, L, p, ΔV ^*, r and build a realization of a network (i.e., W and \(V_n^*\) are known for all n).

We call connection properties at the first order all the direct connections of the neuron in the network contained in the matrix of synaptic weights W (this means the number of sons and fathers (see Section 2.1)). In particular, we do not introduce any information about higher order properties, such as the number of paths between two neurons. All these higher order properties are obtained from W above.

We call first order properties all the neuron properties and all the connection properties at the first order.

We call prediction efficiency the coefficient \(p_e=\frac{\sum_{n}\textrm{correct prediction}_n}{N}\) where correct prediction_n is 1 if the neuron n is a leader α _n > 3 (respectively not a leader α _n ≤ 3) in the simulation and its prediction p _n > 3 (respectively p _n ≤ 3), otherwise: correct prediction_n = 0.

This means a network in which all neurons have exactly the same number of connections: the eight nearest neighbors (unlike Fig. 1). However, the membrane potential firing threshold of the neurons differs.

This fact was already explained in Section 4.2.

All the neurons with the same membrane potential firing threshold, type and number of fathers and sons.

Our model was done to mimic in vitro experiments (Eckmann et al. 2008) in which the inhibitory proportion r is about 20%. Normally 0 ≤ r ≤ 1. So by choosing r = 0 we are in the edge of the parameter space. That is why we pretend that the abnormally high prediction error is a kind of edge effect.

In a Gaussian distribution, which is expected for the components of \(\vec x\), the interval that contains the mean value more or less three times the standard deviation includes 99% of the results.

This means that we need to compute W ² to extract this property.

The time position 0 in Fig. 11 is reserved for the trigger of the burst.

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Title: Leader neurons in leaky integrate and fire neural network simulations
Author: Cyrille Zbinden
Publication date: 01-10-2011
Publisher: Springer US
Published in: Journal of Computational Neuroscience / Issue 2/2011
Print ISSN: 0929-5313
Electronic ISSN: 1573-6873
DOI: https://doi.org/10.1007/s10827-010-0308-6

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