Robustness of Persistent Firing in a minimal recurrent network of Working memory - Дипломная работа

Robustness of Persistent Firing in a minimal recurrent network of Working memory Chapter 1. Introduction From a reductionist point of view, it is trivial that understanding the individual mechanism and interaction of neurons, as the basic constituents of the nerves system, is essential for understanding the brain as a whole. But on the other hand, a great tendency is growing toward applying reduced models in theoretical neuroscience. The biologically realistic population models consisting of thousands of interacting neurons are highly nonlinear and laborious to be analyzed thoroughly, while reduced models can be very simplified yet physiologically plausible frameworks for studying perceptual mechanisms. A relevant example for the application of reduced models, is a study by Gutkin, Laing, Colby, Chow, & Ermentrout (2001), focusing on the role of spike timing asynchrony and synchrony in sustained neural activity. In their paper a simple two-cell circuit were compared with a population model comprising 500 connected neurons, and it was shown that dynamical properties of their proposed reduced model is sensibly analogous to that of the population model. As another instance, we can point to the popular model of decision making by Wang (2002) consisting of 2000 neurons, which was later replaced by a successful simplified version of two coupled neurons, capable of reproducing the same properties of the original model (Wong & Wang, 2006). Except from complexities of large scale population models that should be tackled, the individual single cell models must be also relatively simple as well as being biologically credible. Both electrophysiological and dynamical properties of neurons are important for information processing in the brain (Izhikevich, 2007) and numerous computational models of neurons have been introduced to feature neural dynamics with different levels of complexity. Physiologically detailed computational models of cells, such as the famous Hodgkin-Huxley (HH) model (Hodgkin & Huxley, 1952) or the detailed compartmental models (Brunel, Hakim, & Richardson, 2014; Almog, & Korngreen, 2016) are too complicated to be analyzed mathematically, while relatively simple canonical models are advantageous due to their generic nature, retaining many important features of a whole family. Minimal models of cells due to definition of Izhikevich, (2007) are those having minimal sets of currents that enable them to generate action potentials. (1) In this equation, x characterizes the membrane voltage of the neuron and Iext is a parameter determining the activity mode. On the one hand, a negative Iext introduces two real equilibriums to the system, . For simplicity these fixed points are referred to ‘a’ and ‘-a’ in this work. The negative root ‘-a’ is the stable node that plays the role of a rest state, while the positive root, ‘a’, is a saddle node, playing the role of a threshold for spike initiation. Voltage trajectory always converges to the rest state unless it can cross the Threshold. Any x value beyond the threshold will grow to create an action potential. It will quickly hit Vpeak, regresses to a reset parameter, Vreset, and eventually tends toward the rest state again. In fact, existence of these two real roots keeps the cell in an ‘Excitable’ mode, which means that the neuron is disposed to fire a single spike by a sufficiently large external pulse, but it doesn’t exhibit regular pattern of firing. On the other hand, a neuron with a positive Iext, loses its real equilibriums and launch into a ‘Periodic’ mode of activity. Thus, Iext = 0 is a bifurcation point for this model. In figure 2 the QIF function is indicated for three values of Iext, below, above, and at the bifurcation point. This figure illustrates how increasing the external current from panel a to c leads to occurrence of saddle node bifurcation and evanescence of the fixed points. Figure 2. dX/dT versus X and fixed points of the QIF model. a) Excitable neuron. b) Saddle node bifurcation, c) Periodic firing mode (Izhikevich, 2007). 1.3 Coupled QIF neurons Gutkin, Jost, and Tuckwell, (2008a) proposed that a pair of coupled QIF neurons are able to produce self-Sustained neural activity resembling the persistent activity in a delay period of a working memory task in Prefrontal Cortex (PFC) neurons. Following that study, a more recent publication by Novikov and Gutkin (2016) have investigated the dynamical structure of the same network with fine mathematical details. As mentioned earlier, in this work, we will investigate the dynamical features of a minimal network with time dependent synapses, in reaction to both excitatory and inhibitory pulses. We will indicate that for such non-autonomous systems robustness varies with respect to the different phases of the limit cycle. Finding the attraction basins and regions of stability is not straight forward for this system, because everything depends on time. However, by numerical compu