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Fig. 9 | The Journal of Mathematical Neuroscience (JMN)

Fig. 9

From: A Formalism for Evaluating Analytically the Cross-Correlation Structure of a Firing-Rate Network Model

Fig. 9

Correlation at \(t=10\) and for \(\sigma=0.1\), as a function of the number of incoming connections M, in the case of the circulant connectivity matrix \(\operatorname{Ci}_{N} (1,2,\ldots,\xi )\) (left) and of the block-circulant matrix \(\mathcal{BC}_{2,{N}/{2}} (\mathcal{M}_{0},\mathcal {M}_{1} )\), with \(\mathcal{M}_{0}=\mathcal{M}_{1}-1=2\xi-H (\xi-\frac {N}{4}+1 )\) (right). The number of incoming connections is, respectively, \(M=2\xi -H (\xi-\frac{N}{2}+1 )\) with \(1\leq\xi\leq \lfloor\frac{N}{2} \rfloor\), and \(M=1+4\xi-2H (\xi-\frac{N}{4}+1 )\) with \(1\leq\xi\leq \lfloor\frac{N}{4} \rfloor\). These results have been obtained by using Eq. (6.7) with \(C^{ (0 )}=C^{ (1 )}=C^{ (2 )}=0\) (while all the remaining parameters are those of Table 1). The figure shows that correlation does not go to zero in the thermodynamic limit (absence of propagation of chaos) if \(\lim_{N\rightarrow \infty}M\) is finite, namely if the network is sufficiently sparse

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