ISSN electrónico 2011—7531 ISSN impreso 0120—5552 Vol. 34, N° 1, 2018 Fecha de recepción: 15 de julio de 2017 Fecha de aceptación: 30 de agosto de 2017 DOI: http://dx.doi.org/10.14482/sun.34.1.9736

ARTÍCULO DE INVESTIGACIÓN / RESEARCH REPORT

http://dx.doi.org/10.14482/sun.34.1.9736

Oscar E Hernández¹, Eduardo E Zurek², David Vera³, Alfonso Cepeda—Emiliani⁴

¹ Departamento de Química y Biología, Universidad del Norte, km 5 vía a Puerto Colombia, Barranquilla,Colombia. Orcid: 0000—0003—4476—0055

² Departamento de ingeniería de sistemas, Universidad del Norte, km 5 vía a Puerto Colombia, Barranquilla, Colombia. Orcid: 0000—0002—9816—6863

³ Departamento de Ciencias Naturales y exactas, Universidad de la Costa, Calle 58 N° 55—66, Barranquilla, Colombia. Orcid: 0000—0003—3955—5452

⁴ Departamento de Medicina, Universidad del Norte, km 5 vía a Puerto Colombia, Barranquilla, Colombia. Orcid: 0000—0002—5269—7810

Correspondencia: Oscar Emilio Hernández Bustos, universidad del norte, km 5 vía puerto Colombia. Barranquilla,Colombia. Tel. +5753509509 ext 4284 ohernandezb@uninorte.edu.co

Fecha de recepción: 15 de julio de 2017
Fecha de aceptación: 30 de agosto de 2017

Abstract

Objective: To show the relation between the four parameters associated to bursting discharges of the thalamic reticular cells (TRNn): the maximum firing frequency (f_max) and the temperature at which it occurs (T_fmax), the range of temperatures defined as the full width at half maximum (AT_h) and the maximum specific low threshold calcium conductance (G_T).

Materials and Methods: In order to simulate the TRNn bursting activity, a computational simulation model was implemented using the NEURON software, which incorporates morphological and electrophysiological data, and stimuli properties closely related to reality. Results: It was found that there are nonlinear relations between the parameters. The f

max frequency follows a quadratic growth with temperature and tends asymptotically towards a limit value with the maximum calcium conductance. In the same manner, ATh increases until reaching a limit value as function of f_max and G_T. However, the increment per frequency unit is bigger than the increment per conductance unit.

Conclusions: Four equations were obtained that model the relations between the parameters associated to bursting discharges of the TRNn in rats and other neurons with similar characteristics in different animal species.

Keywords: TRN neuron; Computational simulation; Firing frequency; Temperature; Burst firing

Resumen

Objetivo: Mostrar la relación entre los cuatro parámetros asociados a las descargas en ráfaga de las neuronas del núcleo reticular del tálamo (TRNn): la frecuencia máxima de descarga (f_max) y la temperatura a la cual se produce (T_fmax), el rango de temperaturas definido como ancho a media altura (AT_h) y la conductancia máxima de calcio de bajo umbral (G_T).

Materiales y métodos: Para simular las descargas en ráfaga de las TRNn se implementó un modelo de simulación computacional usando el software NEURON, que incorpora datos morfológicos, electrofisiologicos y las propiedades de los estímulos en estrecha relación con la realidad.

Resultados: Se encontraron relaciones no lineales entre los parámetros. La frecuencia f max crece de forma cuadrática con la temperatura y tiende asintóticamente a un valor límite con la conductancia. Así mismo, AT_h también se incrementan hasta alcanzar un valor límite en función de f_max y G_T. No obstante, es mayor el incremento por cada unidad de frecuencia que por cada unidad de conductancia.

Conclusiones: Se obtuvieron cuatro ecuaciones que modelan las relaciones entre los parámetros asociados a las descargas en ráfaga de las neuronas TRN en ratas y otras neuronas con características similares en diferentes especies animales.

Palabras clave: Neurona del TRN; Simulación computacional; Frecuencia de disparo; Temperatura; Descarga en ráfaga

INTRODUCTION

Thalamic reticular nucleus neurons (TRNn) respond to inputs in two different activity modes called burst and tonic firing [1, 2, 3, 4]. Burst mode is caused by dendritic activation of low threshold calcium currents (I_T) during inhibitory inputs, and this is mediated by T—type calcium channels with the properties: low unitary conductance, fast inactivation, slow activation kinetics, negative steady—state inactivation at resting potential and resistant to dihydropyridme (DHP) [5, 6]. The tonic mode is generated in response to excitatory inputs. Current I_T is mediated by a specific low threshold calcium conductance (T—type conductance) and is expressed in various cells, tissues and organs of species with different ranges of internal temperature (ΔT) and maximum values of T—type conductance (G_T). These include nervous tissue, smooth muscle, endocrine glands, sperm, kidneys, and heart [7, 8, 9]. In the lateral geniculate nucleus of the cat, for example, G_T = (8,4 +/— 0,9) nS/cm² and AT = (36,9 — 38,0)°C [10]; in the thalamic ventrobasal nucleus of the rat, G_T = (37600 — 38600) nS/cm² and AT = (34,0 — 36,0)°C [11]; in the dorsal root ganglion of the chick, G_T = (0.020—0.025) nS/cm² and ΔT = (40,8 — 41,6)°C[12]; in the dorsal root ganglion of the mouse, G_T = 0,0072 nS/cm² and 0,0057 nS/cm², and ΔT = (36,5 — 38,0)°C[13]; and in human embryonic kidney 293 cells (HEK 293), G_T = 0.009 nS and ΔT = (36,0 — 37,5)°C[14].

Employing computational simulations, it was previously shown that bursts and tonic firings observed in a thalamic reticular nucleus (TRN) soma could be explained as a product of random synaptic inputs on distal dendrites [15]. Tonic firings are generated by random excitatory stimuli, while two different types of stimuli generate burst firings: inhibitory random stimuli and a combination of inhibitory (from TRNn) and excitatory (from corticothalamic and thalamocortical neurons) random stimuli. Other studies have highlighted the importance of burst discharge parameters such as number and patterning of spikes, firing frequency, and rhythm of firing in transport and neuronal information processing [16, 17, 18, 19, 20, 21, 22]. The burst activity mode has an important effect on information processing in thalamocortical circuits during different physiological states, such as synchronized sleep and anesthesia [23, 24].

Commonly, intracellular recordings of electrical activity are conducted at room temperature and not at physiological temperature [25, 26, 27, 28, 29]. Increases in the opening and closing rates of ion channels when temperature (T) rises 10°C above room temperature are expressed by the factor Q10, which is proportional to the activation energy [30, 31, 32, 33]. However, it has been found that the effects of temperature on T—type Ca⁺² channels are nonlinear [34, 35]. Computational simulation offers multiple alternatives, not only for modifying electrical neuronal conduction properties, but also environmental conditions such as temperature [36, 37, 38, 39, 40]. Accordingly, employing computational simulations, this study shows a nonlinear relation between the maximum burst firing frequency (f_max) observed in TRNn and the temperature at which it is produced on the one hand, and G_T, on the other. For a given G_T, the range of temperatures at which TRNn responds in burst mode also increases nonlinearly (exponential growth), both with its f_max and its G_T conductance.

MATERIALS AND METHODS

In order to develop a realistic simulation, the use of a morphology (neuronal geometry) obtained from a neuronal reconstruction with real dimensions was considered, the application of stimulus that takes into account the randomness of the amplitudes and durations of both inhibitory and excitatory stimuli originating in other TRNn and in collaterals of thalamocortical and corticothalamic neurons, respectively; and the biophysics that are incorporated by the model allow for reproducing different types of neuronal responses and their properties.

Then, the activity of TRNn was simulated using a computational tool, developed in NEURON simulation environment [41, 42], that incorporated morphological and electrophysiological data, and the stimuli properties. The following parts of this section describe in detail each one of these aspects and the methodology for analyzing burst responses of TRNn.

MORPHOLOGICAL DATA

The geometrical data (Table 1) were obtained from a neuronal reconstruction of an intact TRN cell with 80 compartments (figure 1A), from the ventrobasal region of a rat [43]. The data files were downloaded from the web page of the SenseLab Project [44].

ELECTROPHYSIOLOGICAL MODEL

The biophysical model with high distal density of g_T (Table 2) corresponds to a TRN cell from the ventrobasal region of a rat and it was obtained from [43]. The data files were downloaded from reference the web page of the SenseLab Project [44]. Tables 3—6 include a summary of the main formulas used in reference [44] to develop the algorithms that model the biophysics of neurons TRNn.

STIMULI PROPERTIES

The typical bursts in the soma of the TRNn can be generated by joint trains of inhibitory and excitatory random stimuli in the distal dendrites [15]. TRNns are continuously bombarded by excitatory stimuli from the collaterals of thalamocortical and corticothalamic neurons [45, 46], and by inhibitory stimuli of other TRNn [47]. A dendrite can be stimulated by one or more inhibitory postsynaptic potentials (IPSP) and excitatory postsynaptic potentials (EPSP) resulting in a temporal and spatial summation into the same dendrite [48, 49]. Thus, each dendrite can contribute with a depolarizing or hyperpolarizing potential directed toward the soma. The amplitude of this potential depends on the number of EPSPs and IPSPs that stimulate the dendrite, and in consequence, can be considered as a random event.

Then, the characteristics of all trains of stimuli were randomized following a uniform distribution and were estimated from the analysis of the results presented in references [47, 15]. The amplitude of each stimulus (square pulses) was in the range [—0.25, 0.1043] nA, the duration of each stimulus was in the range [0, 1.0507] ms and the start time varied within [80, 120] ms. 25 pulses were applied with delays in the range from 0 to 12.4000 ms, obtaining a frequency of 0.1553 kHz.

The simulation presented in this paper evaluates 60 distal dendrites, and a different train of random stimuli was applied to each one of them. The trains of stimuli have the same statistical properties, but each train has a different random seed. The amplitude and duration of the stimuli follow a uniform distribution between a lower limit(l_lower) and an
upper limit (l_upper). Then, if l_lower < 0 and l_upper >
0, the EPSP rate (EPSP rate) can be estimated as:

METHODOLOGY FOR ANALYZING BURST RESPONSES OF TRNN

From the above parameters and using a NEURON simulation environment [41, 42], it follows that an applied alternative methodology could allow for an approximation of the way the TRNn, or any other type of neuron, respond to changes in body temperature. It could determine the relationship between the burst mode response frequency (f) of the TRN neuronal model (TRNnm) and physiological temperature (T) with specific G_T values (figure 2A).

This geometrical model was obtained from reference [44]. The cell area using the procedure described in [52]

This biophysical model was obtained from reference [44]. G_NaHH2: Maximum specific Sodium conductance; G_KHH2: Maximum specific Potassium conductance; G_T: Maximum specific low threshold calcium conductance (for the distal dendrites, this values varies in a range [5.8, 10] x10^—4 S/cm²); G_pass: Specific leak conductance; E_Na: Sodium equilibrium potential; E_K: Potassium equilibrium potential; E_T: Calcium equilibrium potential; E_pass: Leak equilibrium potential; R_a: Axoplasmic resistance; C_m: Specific membrane capacitance; K_d: Dissociation constant (assuming a Michaelis—Menten approximation for a simple model of ATPase calcium pump). The dashed line indicates the absence of the property.

RESULTS

The results of this work are focused, first, on the analysis of the relationships between f in burst max mode and the corresponding temperatures (see Figure 2B). Second, the results are focused on f _max and the G_T at which it was obtained (see Figure 2C). Third, they are centered on the relationship between ΔT. and the corresponding f values h _max (see Figure 3A), and the relationship between ΔT_h and G_T (see Figure 3B). ΔT_h is a parameter proportional and representative of the temperature range at which TRNn can respond in burst mode (see Figure 3A).

Figure 2A shows the relationship between the burst mode response frequency (f) of the TRNnm and physiological temperature (T) for G_T = 8 S/ cm², and for a temperature range in which burst discharges were observed. T_h1 and T_h2 are the temperatures at which the TRNnm can respond in burst mode with a frequency equal to half the maximum frequency (i.e. f = f_max/2). ΔT_ft = T — T is full width at half maximum. T is theh2 h1 f_max temperature at which the TRNnm fires in burst mode at maximum frequency. It was found that for any G_T value in the range of [5,8,10] S/ cm², the curve fitted to points (f,T) is always biased to the left (negative asymmetry) in a maximum temperature range [12,44]°C, and the fitted equations were degree 10 polynomials. All the parameters that generated the results of this work were based on the development of graphics similar to Figure 2A (f vs. T) and for different G_T values in the aforementioned range.

other values of ^G_T^, the equation fitting was similar. T_h1 and T_h2 are the temperatures at which the TR—Nnm responds in burst mode with a frequency equal to half the maximum frequency (f = f_mox/2). Δ_Th = T_h2 — T_h1. Tf_max is the temperature at which the TRNnm fires in burst mode at maximum frequency. For any value of ^G_T^, this curve always shows a negative asymmetry. B) Relationships between f_max and T . The points (f ,T ) on the solid line were fitted to a growth quadratic equation (Eq. 1). C) fmax max fmax Nonlinear relation between f and G . The points (f ,G ) on the solid line were fitted to a growth max T max T exponential equation. The dashed lines in B) and C) represent the first derivative of equations 1 and 2, respectively.

The solid line in Figure 2B shows the relation between f and T and its points fitted to max fmax a growth quadratic equation (Eq. 1, with R² = 0.99623).

The dashed line in Figure 2B represents the first derivative of equation 1 (see eq. 2). It shows that as temperature rises, the maximun burst mode firing frequency per unit temperature of the TRNnm increases in linear and growing form. Thus, for example, at T = 37°C, f increases by 1 kHz for each temperature max increase of 1°C; and at T = 40°C, f increases max by 1354 kHz for each temperature increase of 1°C. 2.485x10^—3 Hz/°C/°C represents the slope of the line and shows the steady increase of dfmax/dT per unit temperature increase, i.e., df /dT increases by 2.485 x10^—3Hz/°C for max each temperature increase of 1°C.

The points (f_max, G_T) were also fitted to a growth curve, but in this case following an exponential relation (eq. 3, with R² = 0.99901), as shown by Figure 2C. 1.0529 kHz represents the maximum value that fmax can reach no matter how high the G_T is. In equation 3, it can be seen that this saturation value is obtained for very large values of G_T (if G_T approaches infinity). However, this TRNnm can only respond with typical burst activity for the range of conductances [5.8, 10] S/cm².

The dashed line in Figure 2C corresponds to the first derivative of equation 3 (see eq. 4). It shows that as G_T increases, the maximum frequency changes observed in the TRNnm decrease exponentially, i.e., df_max/dG_T decreases if G_T increases, and tends to zero if G_T tends to infinity. For example, for G_T = 6x10^—4 S/cm², df_max/dG_T decreases by 0.11 kHz/(1x10^—4 S/ cm²) for each unit increase (1x10^—4 S/cm²) in maximum Ca⁺² conductance. For G_T = 8x10^—4 S/cm², f_max decreases by 0.08 kHz/(1x10^—4 S/ cm²) for each G_T increase of 1x10^—4 S/cm².

Figure 3A shows the relationship between ΔT_h and G_t The points (ΔT_h, G_t) were fitted to a growth exponential equation (eq. 5, with R²= 0.97859). with the maximum value ΔT_h =15.321°C. This was expected since, despite being different equations, the same ΔT_h maximum values should be observed.

Figure 3B shows the relationship between ΔT_h and f . The points (ΔT, f ) were fitted to a max h max growth exponential equation (eq. 6, with R² = 0.96501). 15.321°C represents the maximum value that ΔT_h can reach independently of the^fmax value.

This saturation value occurs if f tends to max infinity.

From equations 5 and 6, it can be observed that f = 0.086 kHz and G = 6.37 S/cm² max Trepresent the values at which ΔT_h reaches a value equal to 63% of its maximum possible value (15.321°C), i.e. ΔT_h

DISCUSSION

The physiological temperature of mammals may vary depending on different normal or pathological causes [50]. However, the electrical response of the different types of neurons to normal and pathological temperature variations is not fully understood [51]. This study presents a novel methodological alternative based on computational simulations, which allows for predicting the way different types of neurons respond to changes in body temperature. The following parameters were defined for each frequency (f) versus temperature (T) graphic with specific G_T values: full width at half maximum (ΔT_h) as a measure of the temperature range at which a neuron responds in burst mode, f , and the max temperature at which this frequency occurs (T_fmax). Next, based on the development of this same graph for different G_T values, graphics 2B, 2C, 3A and 3B were obtained.

The results of this study are a faithful approximation to reality due to four main reasons. First, the use of a morphology (neuronal geometry) obtained from a neuronal reconstruction (intact TRN cell with 80 compartments) that entails real dimensions. Second, the biophysics incorporated to the model allows reproducing different types of neuronal responses and their properties. Third, the applied stimulus takes into account the randomness of the amplitudes and durations of both inhibitory and excitatory stimuli originating in other TRNn and in collaterals of thalamocortical and corticothalamic neurons, respectively. Fourth, consistent with previous experimental studies, this study found that the effects of temperature changes in T—type Ca⁺² channels are nonlinear (see Figures 2, 3 and equations 1, 3, 5 and 6).

All results obtained in this work correspond to a specific TRNnm. Nonetheless, T— type Ca⁺² channels cause the observed bursting activity. The following observations highlight some of the most relevant results of this study and their electrophysiological implications.

Figure 2B and Equation 1 show that for a constant conductance, increases in physiological temperature cause an increase in the maximum firing frequency and the effect is greater for higher temperature values. Although the quadratic growth model shown in figure 2B and equation 1 suggests that there is no f limit for any temperature increase, max

figure 2C and equation 3 show that for high G_T values, maximum firing frequency of the TRNnm always tends to an f limit. For the max model used in this work, the limit tends to fmax= 1.0529 kHz

The results in Figure 2 show that the development of computational neuronal models incorporating reality—adjusted biophysics and geometry, can contribute to predicting the behavior of a neuron for different T—type calcium conductances and physiological temperatures. The curves fitted to Figures 2B and 2C (see equations 1 and 3, respectively) have determination coefficients very close to a unit (R²=0.996), which shows the reliability of the model.

Figure 3A and equation 5 show that if TRNn display bursting activity with high f_max values, then they can display such responses through a wide range of temperatures. In addition, Figure 3B and equation 6 show that if bursting activity caused by T—type calcium conductances for high G_T values is observed in a neuron similar to TRNn, then it can exhibit this kind of response through a wide range of temperatures. Specifically, the proper functioning of T—type Ca⁺² channels after broad physiological temperature variations (increase or decrease) depends on G_T. From Equations 5 and 6, it can also be inferred that if G_T = 6.37 S/cm² and bursting activity is observed at a maximum frequency of 0.086 kHz, then this neuron can exhibit the same type of activity through a range of temperatures Al. > 9.65°C.

CONCLUSIONS

Based on a computational TRNnm that responds in burst mode due to the application of the same type of random stimulus (see Figure 1) at different points of the distal dendrites, it can be concluded that:

First, for constant G_T, the maximum firing frequency of TRNn varies proportionally and in quadratic relation to the temperature at which it occurs (see Equation 1). This equation shows that for the same specific T—type calcium conductance, increases in physiological temperature cause an increase in the maximum firing frequency and the effect is greater for increments in temperature values.

Second, the maximum frequency at which TRNn respond in burst mode varies proportionally and in exponential growth relation to the maximum Ca⁺² conductance (see Equation 3). This equation shows that for increments in the values of G_T, the maximum firing frequency always tends to an fmax limit.

Third, ΔT_h is a representative value of the temperature range (ΔT) at which specific TRNn respond in burst mode, and its relation to the maximum firing frequency in burst mode (ΔT_ft vs. fmax ) was fitted to an exponential growth (see Equation 6). This equation shows that if specific TRNn display bursting activity with fmax values, they can also display this type of behavior through a wide range of temperatures.

Finally, and similar to the above relationship (ΔTh vs. fmax), ΔTh vs. GT was fitted to an exponential growth equation (see Equation 5). This equation shows that the proper functioning of T—type Ca+2 channels after broad physiological temperature variations depends on GT.

Conflict of interest: none.

Financing: own resources.

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