Chaos in memory function of sleep: A nonlinear dynamical analysis in thalamocortical study

Highlights

• A complete analysis of the nonlinear biological thalamocortical model for up-down oscillations and fast-slow spindles.

• The investigation of the bifurcation diagram, attractors, and power spectral for the thalamocortical model.

• By increasing the fast-slow spindles, the thalamocortical network shows chaotic behavior to improve memory process in sleep.

Abstract

Studying the dynamical behaviors of neuronal models may help in better understanding of real nervous system. In addition, it can help researchers to understand some specific phenomena in neuronal system. The thalamocortical network is made of neurons in the thalamus and cortex. In it, the memory function is consolidated in sleep by creating up and down state oscillations (1 Hz) and fast (13–17 Hz) – slow (8–12 Hz) spindles. Recently, a nonlinear biological model for up-down oscillations and fast-slow spindles of the thalamocortical network has been proposed. In this research, the power spectral for the fast-slow spindle of the model is extracted. Dynamical properties of the model, such as the bifurcation diagrams, and attractors are investigated. The results show that the variation of the synaptic power between the excitatory neurons of the cortex and the reticular neurons in the thalamus changes the spindles' activity. According to previous experimental findings, it is an essential rule for consolidating the memory function during sleep. It is also pointed out that when the fast-slow spindles of the brain increase, the dynamics of the thalamocortical system tend to chaos.

Introduction

Memory consists of three stages, encoding, consolidation, and retrieval (Anderson and Green, 2001). Encoding is developed based on the perception of input information. It may first disappear during the waking period due to time interferences. Newly encoded information during wakefulness is gradually transformed into long-lasting memory traces during subsequent sleep. Also, they are consolidated during the unique process of sleep. Experimental evidence in the EEG signal has shown that the transition from wakefulness to sleep is caused by various factors such as circadian and hemostatic factors, the interaction between active sleep neurons in the hypothalamus, and variation of slow-wave sleep (SWS) oscillations (Anderson and Hanslmayr, 2014, Andrillon et al., 2011, Babadi et al., 2012). The sleep process consists of REM (Rapid eye movement) and NREM (Non-rapid eye movement). NREM sleep has four stages: N1, N2, N3, N4. Stages 3 and 4 of NREM sleep are known as SWS oscillations (Barrio et al., 2014). The sleep stages are presented in part (a) of Fig. 1. In the SWS stage, the thalamocortical network plays an essential role in the memory consolidation process (Blaskovich et al., 2017). The thalamocortical networks are made of neurons in the thalamus and cortex (Bliss et al., 2003). Most of the oscillations in SWS include up-down state (UDS) oscillation, sleep spindles, and sharp ripple hippocampal waves (part (b) of Fig. 1) (Barrio et al., 2014). The SWS oscillations are based on neural currents and synapse connections (Brette and Gerstner, 2005). The hippocampus is associated with short-term memory recording, while the neocortex is associated with long-term memory recording. The interaction between the slow cortical oscillations, the sharp ripple waves of the hippocampus, and the fast-slow spindle (FSS) of the thalamus facilitate the connection of the hippocampus and the neocortex. Therefore, short-term memory is transferred from the hippocampus to long-term memory in the neocortex. Thus, the sleep spindles act as an intermediate component for the communication between the neocortex and the thalamus in the process of memory consolidation (Buzsaki, 2006). The anatomical position of the cortex, hippocampus, and thalamus are shown in part (c) of Fig. 1.

Gais et al. have shown that the density of sleep spindles is raised after learning. This fact increases the peaks of the first cycle of night sleep (the first 90 min) (Cacioppo et al., 2007). Some researchers have demonstrated the correlation between memory consolidation and sleep spindles (Chausson et al., 2013). The consolidation of memory during sleep is carried out by repeating the activation of the neural networks that were previously involved in storing relevant information (Buzsaki, 2006). The envelope of the curve of a sleeping spindle, from the beginning to the middle, is incremental, and then to the end, it has a decreasing trend. That is why it is called a sleep spindle. The sleep spindle consists of two parts, the fast spindle with frequencies 13–17 Hz in the upper part of the brain and the slow spindle with frequencies 8–12 Hz in the parietal and central regions of the brain (Che et al., 2009). Studies on the EEG signal show the vital and active role of the sleep spindle in memory consolidation (Conklin and Eliasmith, 2005).

The cerebral cortex is a group of neurons that cover the entire surface of the brain as the mantle (Contreras et al., 1997, Destexhe, 2009, Diekelmann and Born, 2010, Ditlevsen and Lansky, 2005). Two components of excitatory (EX) and inhibitory (IN) neurons are used to model the dynamics of cortical systems (Edin et al., 2007). The axon terminal of a neuron is connected to the dendrite of the other neurons by a synaptic connection. If the primary neuron is excitatory, then the signal has a depolarizing property, known as excitatory postsynaptic potential (EPSP). If the primary neuron is inhibitory, the signal has a polarizing property, known as inhibitory postsynaptic potential (IPSP) (Edin et al., 2007). The thalamus is located in a central position of the brain, above the brainstem, and below the cortex (Fenton and Cherry, 2008). It is divided into two parts: The first one is the posterior thalamus, which contains the thalamocortical (TC) neurons, made of relay cells and associated with the neocortex. The second one is the ventricular thalamus that contains the reticular (RE) neurons (Fenton and Cherry, 2008). TC neurons are excitatory and use glutamate as a neurotransmitter. These flows are used for a series of oscillating activities (Fischer et al., 2011). RE neurons are inhibitory and have neurotransmitters of the gamma-aminobutyric acid (GABA) type (Foroutannia et al., 2020). The thalamocortical network is the source of various oscillating activities with different mechanisms.

Mathematical modeling, from the dynamical systems point of view, has been a hot topic to study a neuron or set of neurons' activities (Gais et al., 2002, Galves and Löcherbach, 2013). Different neuronal models are classified based on electrical input–output membrane voltage models (such as Hodgkin–Huxley (Gerstner, 2000), leaky integrate-and-fire (Gerstner et al., 2014), fractional-order leaky integrate-and-fire (Ghasemi et al., 2021), exponential integrate-and-fire (Ghasemi and Foroutannia, 2019), adaptive exponential integrate-and-fire (Ghasemi et al., 2021), stochastic models of membrane voltage and spike timing models (such as noisy input (Ghorbani et al., 2012), spike response (Gorur et al., 2002), Galves–Löcherbach (Haghighi and Markazi, 2017), didactic toy models of membrane voltage (such as FitzHugh–Nagumo (Hashemi et al., 2019), Morris–Lecar (Hill and Tononi, 2005), Hindmarsh–Rose (Hindmarsh and Rose, 1984), theta model and quadratic integrate-and-fire (Hodgkin and Huxley, 1952). Zhang et al. have shown that increasing the electrical flow in the synaptic connection of the axon of a neuron with its dendrite causes an increase in the firing rate, which strengthens memory (Hou et al., 2020). Lv et al. considered magnetic flux to describe the effect of an electromagnetic field on neuronal activity. It was stated that the field connection between neurons could alter the collective behavior of electrical activities (Izhikevich, 2007). The electrical activity of neurons under the depolarization field was studied in (Jing et al., 2004). In Koch and Segev (1998), a pattern of the neuronal set was designed as a two-dimensional neural network that examines the dynamical behaviors of the system under the influence of magnetic radiation. The effect of various noise types on the FitzHugh-Nagumo (FHN) neuronal model was studied in Krugers et al. (2011). In Kumar et al. (2013), the effect of energy on the signal transmission of neurons was investigated. The discharge mode of a neuron was discussed in Liu et al. (2019). The state of the chimera in neuronal and biological models was investigated in Loomis et al. (1935). Haghighi et al. analyzed the dynamical properties of the thalamocortical model for the diagnosis of epilepsy (Lv et al., 2019).

In this research, a simple cognitive thalamocortical network based on the theory of dynamical systems is analyzed. This model shows the up-down oscillations and sleep spindles. The cortical network consists of excitatory and inhibitory neurons. On the other hand, the thalamus network consists of TC and RE neurons. The studied model includes a thalamus network (Majhi et al., 2017), a cortex network (Majhi et al., 2019), and the combination of the two networks of thalamus and cortex, which was introduced in (Marzano et al., 2013). In the paper, the dynamical properties of the thalamocortical network are investigated, and the effect of synaptic connections is discussed. Also, the effect of FSSs on the consolidating of memory during sleep is studied.