Synfire Chains in a Balanced Network

We investigate the formation of ordered spatiotemporal activations of pools of neurons in synfire chains (SFC) within a balanced network of excitatory and inhibitory neurons, both by simulations on integrate-andfire (IF) neurons and by analytic tools. Using a suitable matrix of synaptic connections, we show that the results depend on the ratio r between the size of an individual pool of neurons and the total excitatory input of a neuron. In our simulations of 10,000 IF neurons we obtain an asynchronous-irregular firing mode, which does not sustain a traveling wave of SFC activity. Our analysis shows that the latter may be expected to exist in larger networks in which very small r values can be realized. 1 Summary A sparsely connected network, where each neuron gets as much inhibition as excitation is called balanced network [6]. We construct such networks that have excitatory-excitatory synaptic connections that consist of ordered links between pools of neurons, such that if neuronal activity would follow these links a synfire chain (SFC) [1] would be formed. We study the question whether this is indeed the case in a balanced IF network that contains also random inhibitory connections, and stochastic external input, allowing it to operate in an asynchronous-irregular (AI) mode [3]. Our analytic studies are performed in a binary model that helps us gain insight into the issues involved in the simulations. Previous studies of SFC [2] [5] were performed in networks with simplified temporal structure, whose activity followed the designed structure of the excitatory synaptic matrix. Our IF network has continuous temporal dynamics, it operates under balanced excitatory-inhibitory conditions and is influenced by stochastic inputs. Hence it is not obvious that the designed excitatory synaptic structure will activate a SFC. The individual neuron is allowed to receive a relatively large input, much larger than a single pool of a SFC would contribute. Nonetheless we aim to achieve a situation in which the firing of a single pool will drive a synfire chain. Our analysis emphasizes the importance of the ratio r=w/K, between the width of the pool w, and the total number of excitatory synapses of each neuron K. We show that the correlation of any pair of neurons in a pool strongly depends on r in a sigmoidal fashion. This is to be expected from any SFC analysis [4], since the higher the ratio r, the higher will be the probability of all neurons in a pool to fire, if driven by a previous pool. The correlation between pairs of neurons increases along a SFC, until it reaches a fixed point, whose value is highly sensitive to the value of r. This result is displayed in Fig. 1. Most of our analytic results, derived in the binary model, also hold qualitatively for the superposition of links model introduced by Bienenstock [2]. 0 2 4 6 8 10 12 14 16 18 20 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 r = w/K in units of 1/sqrt(K) Fi xe dpo in t c or re la tio n Fig. 1. The correlation of two neurons belonging to the same pool of a SFC in a binary model with K=2,000 excitatory inputs, as function of r. Our simulations were carried out on a balanced IF network with 10,000 excitatory neurons and 2,500 inhibitory ones. The network is exactly the same as in [3], except that the connectivity within the excitatory population is not purely random but has a SFC-type structure. We obtained an asynchronousirregular phase, as shown in Fig. 2. The latter is sensitive to the value of r, as can be seen here. We were unable, so far, to obtain a traveling wave of activity of a SFC. We argue that the reason for that is that within the AI regime, the value of w is too small to ignite a SFC. When trying to increase it in the right direction we run into r values that correspond to the upper branch of the sigmoid in Fig. 1. Then the high correlation among all neurons in a pool cause spontaneous emergence of pulse packets. This happens independently to all the pools in the network, driving the network to a global oscillatory mode. We expect stable SFC to occur only for the lower branch of the sigmoid, i.e. for very small r values. To have both low r values and high enough w we have to work with a much larger network. Thus, although not being able to establish SFC in our IF model yet, we conclude that it is not unreasonable to expect it to emerge in a balanced network of 100,000 neurons. 0 50 100 150 200 250 300 350 400 450 500 0 0.005 0.01 0.015 0.02 0.025 0.03 0.035 0.04 po pu la tio n ac tiv ity T ime (ms) pool width = 94