Input Response of Neural Network Model with Lognormally Distributed Synaptic Weights Yoshihiro Nagano 1 , Ryo Karakida 1,2 , Norifumi Watanabe 3 , Atsushi Aoyama 4 , and Masato Okada 1,5+ 1 Graduate School of Frontier Sciences, The University of Tokyo, Kashiwa, Chiba 277-8561, Japan 2 Research Fellow of the Japan Society for the Promotion of Science, Chiyoda, Tokyo 102-0083, Japan 3 School of Computer Science, Tokyo University of Technology, Hachioji, Tokyo 192-0914, Japan 4 Faculty of Environment and Information Studies, Keio University, Fujisawa, Kanagawa 252-0882, Japan 5 RIKEN Brain Science Institute, Wako, Saitama 351-0198, Japan (Received February 19, 2016; accepted May 9, 2016; published online June 6, 2016) Neural assemblies in the cortical microcircuit can sustain irregular spiking activity without external inputs. On the other hand, neurons exhibit rich evoked activities driven by sensory stimulus, and both activities are reported to contribute to cognitive functions. We studied the external input response of the neural network model with lognormally distributed synaptic weights. We show that the model can achieve irregular spontaneous activity and population oscillation depending on the presence of external input. The firing rate distribution was maintained for the external input, and the order of firing rates in evoked activity reflected that in spontaneous activity. Moreover, there were bistable regions in the inhibitory input parameter space. The bimodal membrane potential distribution, which is a characteristic feature of the up-down state, was obtained under such conditions. From these results, we can conclude that the model displays various evoked activities due to the external input and is biologically plausible. 1. Introduction Neural assemblies in the cortical microcircuit can sustain irregular low-firing rate spiking activity without any external inputs. Such firing is referred to as spontaneous activity, and the spikes of which are reported to be irregular, have low firing rate, and are non-periodic. 1–4) Teramae et al. proposed the strong-sparse and weak-dense (SSWD) model, 5) which reproduces such irregular spontaneous activity 1–4,6) by using the physiological findings that the distribution of inter- neuronal synaptic weights follows a lognormal distribu- tion. 7–9) Having a few strong and many weak synapses, the SSWD model can generate noisy spikes internally and maintain spontaneous firings. On the other hand, neurons exhibit rich evoked activities driven by sensory stimulus or top-down signals from the higher cortical area. For instance, it has been reported that visual area V4 shows population oscillation in the range of 40–100 Hz caused by sensory visual stimulus. 10) Such activity is called gamma oscillation. These spontaneous activity and population oscillation are general phenomena observed in various cortical areas. Specifically, both spontaneous activity 6) and population oscillation 10) were reported in V4. For these two phenomena, recent experimental findings suggest that the firing pattern of spontaneous activity constrains that of evoked activity, 11–13) which is known as the sampling hypothesis. 14) Moreover, the gamma-band population oscillation is believed to be linked with high-level cognitive functions such attention, memory, perception, and object-binding problem. 15) For figuring out these experimental results and the theoretical hypothesis, it is important to investigate the neural network model which achieves both spontaneous and evoked activ- ities. There are theoretical works 16,17) investigating the transition between irregular spontaneous activity and evoked population oscillation using a randomly connected neural network model. However, the models proposed in these works cannot sustain irregular spontaneous activity without relying on any external inputs. From biological experimental results, the irregular spontaneous activity is considered to be generated internally. Therefore, the SSWD model is expected to be more biologically plausible, but the input response of the model remains to be uncovered. In this study, we first revealed that the SSWD model, which sustains irregular spontaneous firing without external input, can generate input-driven population oscillation. The oscil- lation frequency was 70–100 Hz, which corresponds to the gamma-band oscillation. Next, we found that the firing rate distribution was maintained for external input, and the order of firing rates in evoked activity strongly correlated with that in spontaneous activity, consistent with the physiological results by Mizuseki et al. 18) Finally, we revealed that when the inhibitory external input was injected to the model, there were bistable regions in the input parameter space. The circuit settled into a firing state or a no-firing state according to the initial firing state. Moreover, bimodal membrane potential distribution was obtained under such conditions, and the model enables the physiologically reported up-down state. 19–23) From these results, we can conclude that the SSWD model displays various evoked activities caused by external input and is biologically plausible. 2. Model 2.1 SSWD model We used the SSWD model proposed by Teramae et al. 5) It is known that the model shows irregular spontaneous firing without externally injected background noise. The dynamics of neurons in the model are formulated as an extension of the leaky integrate-and-fire neuron model. The dynamics of the membrane potential v and excitatory and inhibitory synaptic conductances g E and g I normalized by the membrane capacitance are written as dv dt ¼ 1  mX ðv  V L Þ g E ðv  V E Þ g I ðv  V I Þ; ð1Þ dg X dt ¼ g X  S þ X j G X;j X s j ðt  s j  d j Þ þ G X;ext X s ext ðt  s ext Þ; ð2Þ Journal of the Physical Society of Japan 85, 074001 (2016) http://doi.org/10.7566/JPSJ.85.074001 074001-1 © 2016 The Physical Society of Japan