640 A Frequency Analysis on Calcium T-type Signaling in the Nervous System Alireza Khodaei Hamid Vakilzadian Massimiliano Pierobon Computer Science and Engineering Electrical and Computer Engineering Computer Science and Engineering University of Nebraska-Lincoln University of Nebraska-Lincoln University of Nebraska-Lincoln Lincoln, Nebraska 68588 Lincoln, Nebraska 68588 Lincoln, Nebraska 68588 Email:[email protected] Email:[email protected] Email:[email protected] Abstract—We expressed the importance of frequency charac- mechanisms comprise special types of protein called cal- teristics of T-type calcium signaling in every level of the nervous cium ion channels. The calcium ion channels that regulate system. We also highlighted the need for performing frequency calcium influx are either ligand-gated N-methyl-D-aspartic analysis on calcium signals in order to understand their impact on various functions within the nervous system. We provided the acid (NMDA) channels (NMDA receptors) or voltage-gated necessary analytical and computational frameworks for doing ion channels. Fig.1a illustrates that receiving extracellular power spectral density (PSD) analysis of calcium T-type signals. ligands (e.g., neurotransmitters) from the axon terminals of For this purpose, we modeled a calcium T-type channel gating presynaptic neurons causes the ligand-gated NMDA channels mechanism using a Markov scheme and formulated the calcium in a postsynaptic neuron to regulate calcium influx, while the ionic current with the help of some empirical data. Our analysis shows calcium T-type current fades out rapidly at an exponential voltage-gated calcium channels conduct calcium influx based rate when frequency is increased but decreases almost linearly on the neuron’s membrane potential. Fig.1b also illustrates with depolarization of the membrane. that a calcium channel is basically a transmembrane pore that Keywords—Frequency Analysis, Power Spectral Density, Neu- makes a conductance pathway between the two sides of the rons, Calcium Signaling, Calcium T-type Channels, Calcium neuron membrane that is electrically isolated by a layer of Markov Process, Calcium Monte Carlo Simulation. fatty acid. Please note that the figure shows a basic structure scheme for both ligand-gated and voltage-gated channels. As I. INTRODUCTION we explain later in this section, the voltage-gated T-type Ionized calcium (CAI) is perhaps the most typical signal calcium channels have a bit more complicated structure. transduction element in all of biology. In general, calcium There are two subtypes of voltage-dependent calcium chan- signals (i.e., changes in intracellular calcium concentration nels known to be mainstreams in neural activities, namely, 2+ Ca i over the time) are regarded as universal signals T-type and L-type channels. The T-type calcium channels are that regulate many critical cellular processes, such as gene activated with a small amount of membrane depolarization as expression, proliferation, fertilization, differentiation, contrac- we will discuss in more details in Section.II. In contrast, L- tion, and apoptosis [1] [2]. As for neurons, calcium signals type calcium channels are activated by the high voltage of transduce the membrane potential into actions in every level action potentials and have a longer time of activation. The of the nervous system, including cellular and molecular cir- T-type calcium oscillatory signals—the subject of discussion cuits and systems and cognitive and behavioral levels. For in this paper—are known to regulate thalamocortical rhythms instance, a proper calcium signal initiates a signaling pathway of sleep [7] and performing network synchrony to coordinate that changes synaptic strength through long-term potentiation burst-firing among the neurons [4]. (LTP) and long-term depression (LTD) processes [3], builds Although many studies have been dedicated to identifying memory circuits (engram) in the brain, and eventually brings the roles that calcium signals play in the nervous system, about learning and memory functions in mammalian species few attempts have been made to understand the important [3]. attributes of calcium signaling such as how calcium signals Any irregularity in calcium signaling can cause physio- are modulated by the neuronal intracellular information, and logical and behavioral problems, such as epilepsy seizures what are the underlying mechanisms for the calcium signal [4], elevated anxiety, and impaired memory [5]. Interestingly, modulation. In order to answer such important questions about extracellular calcium signals (i.e., fluctuations of ionized cal- calcium signaling, it is necessary to perform frequency anal- cium outside the neurons) are known to bear intercellular ysis to acquire knowledge about the frequency characteristics information [6]. Therefore, calcium signals can be modulated of calcium signals. by various types of cellular information to regulate a variety of Unfortunately, there is a significant gap in the literature functions in every levels of the nervous system. The neuronal in providing the analytical and computational foundations of calcium signals are actually modulated by the mechanisms calcium signal frequency analysis, even though the literature that control ionized calcium influx into the neurons. These has already recognized the important role of frequency char- 978-1-5090-4767-3/17/$31.00 ©2017 IEEE 641 2+ Ca Ca2+ Cell membrane Fatty acid Tail Voltage-gated Calcium Channel Phosphate Activation head NMDA Receptors Neurotransmiters Subunit (a) Different Types of Calcium Channels (b) Calcium Channel Basic Structure Fig. 1. Calcium Channel Different Types and Basic Structure acteristics in calcium signaling. For instance, [8] discusses II. CALCIUM T-TYPE CHANNEL the ability of calcium signals to operate through frequency The calcium T-type channels are known as low-voltage modulation (FM) to piggyback information across a cell. gating channels. It means that they conduct calcium inux However, no frequency analysis of calcium signal is presented when the neurons membrane is at depolarized subthreshold and hence, valuable information, such as the proper frequency potentials. The threshold here is considered to be a certain range that is used used by the cell to modulate the calcium membrane potential at which the neuron spikes action po- signal, is missing. In [9], an empirical framework is proposed tential(s). Although neurons do not spike action potential to obtain the power spectral density (PSD) of calcium signals in response to subthreshold depolarizing oscillations in their from single-cell-recorded data. Many benefits of analyzing membrane, they generate oscillatory graded potentials. These calcium signals using PSD can be imagined. For example oscillatory graded potentials conserve frequency components in [10], PSD analysis is used to investigate the heart cells of the original membrane oscillations and attenuate while they spontaneous activities happening due to calcium signaling in propagate through the neuron membrane as if they are going cerebellar Purkinje cells. However, obtaining a calcium signal through a frequency-selective load impedance [11] [12]. using PSD with in vitro approaches like the one presented in In fact, individual neurons in complex neuronal assemblies [9] is a rather tedious job with the results varying from trial (e.g., the brain) receive oscillatory subthreshold waves all of to trial. the time due to the properties of both neural circuits and single In this paper, we formulate an analytical framework to neuron cells [13]. The calcium T-type channels transduce obtain neuronal calcium T-type signals using PSD. We also these subthreshold oscillations of the membrane into action perform PSD analysis of calcium T-type signals by performing by modulating calcium inux according to the frequency of a Monte Carlo simulation. We focus our study on the behaviors the oscillations and the local membrane potential. Therefore, and properties of the mechanisms that modulate calcium T- the frequency and amplitude of calcium T-type signals is type signals in the cellular and molecular levels of the nervous correlated to the frequency and magnitude of subthreshold system (i.e., the T-type calcium ion channels in the neurons). oscillations in the nervous system. The dynamics of these mechanisms determine the frequency There are several different structural configurations sug- characteristics of the calcium signal and synthesize the effects gested in the literature for the number of activation and inac- on operations and functions within the upper levels of the tivation gates from which we have adopted the configuration nervous system. Hence, performing a PSD frequency analysis proposed in [14] and [15]. This configuration suggests that the at the cellular and molecular levels helps to identify the aspects structure of a calcium T-type ion channel to be consisted of that impact functions and behaviors within the upper levels of the assembly of two activation subunits as Fig.1b illustrates. the nervous system. For example, knowing the frequency at These activation subunits sometimes are called a subunits as which calcium T-type is highly modulated in a neuron helps to they are made of a proteins. A calcium T-type channel also make hypotheses about the spiking frequency that most likely has an inactivation subunit as we explain later in Section III. modifies synaptic strength and modulate learning and memory. These subunits are sometimes called gates due to the role