Dept. of Neuroscience, Columbia University, New York, NY, 10027, USA; The Edmond and Lily Safra Center for Brain Sciences, Hebrew University. Jerusalem, 91905, Israel

Dept. of Physics and Center for Theoretical Biological Physics, University of California San Diego; Computational Neurobiology Laboratory, The Salk Institute for Biological Studies, La Jolla, CA, 92037, USA

We study the activity of a recurrent neural network consisting of multiple cell groups through the structure of its correlations by showing how the rules that govern the strengths of connections between the different cell groups shape the average autocorrelation found in each group. We derive an analytical expression for the number of independent autocorrelation modes the network can concurrently sustain. Each mode corresponds to a non-zero component of the network’s autocorrelation, when it is projected on a specific set of basis vectors. In a companion abstract we derive a formula for the first mode, and hence the entire network, to become active. When the network is just above the critical point where it becomes active all groups of cells have the same autocorrelation function up to a constant multiplicative factor. We derive here a formula for this multiplicative factor which is in fact the ratio of the average firing rate of each group. As the effective synaptic gain grows a second activity mode appears, the autocorrelation functions of each group have different shapes, and the network becomes doubly chaotic. We generalize this result to understand how many modes of activity can be found in a heterogeneous network based on its connectivity structure. Finally, we use our theory to understand the dynamics of a clustered network where cells from the same group are strongly connected compared to cells from different groups. We show how this structure can lead to a one or more activity modes and interesting switching effects in the identity of the dominant cluster.

To model the heteregenous network we include ^{-1/2}G_{c}_{(}_{i}_{)}_{d}_{(}_{j}_{)}_{i}_{i}_{j}_{=}_{1...N}_{ij}_{j}

_{cd}^{-1}N_{c}_{cd}^{2}_{1}_{1}_{2}

**(A)** For an example network with **(B)** In this example network **(C)** For the two examples networks shown in (A,B) we plotted the trajectory of the autocorrelation vector as a function of the time lag _{1} and V_{2}

**(A)** For an example network with **(B)** In this example network **(C)** For the two examples networks shown in (A,B) we plotted the trajectory of the autocorrelation vector as a function of the time lag _{1} and V_{2}