Effect of temperature in bursting of thalamic reticular cells Efecto de la temperature en las descargas en ráfaga de la células del núcleo reticular del tálamo

Objective: To show the relation between the four parameters associated to bursting discharges of the thalamic reticular cells (TRNn): the maximum firing frequency (fmax) and the temperature at which it occurs (Tfmax), the range of temperatures defined as the full width at half maximum (∆Th) and the maximum specific low threshold calcium conductance (GT). 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 fmax frequency follows a quadratic growth with temperature and tends asymptotically towards a limit value with the maximum calcium conductance. In the same manner, ∆Th increases until reaching a limit value as function of fmax and GT. 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.

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 incorpo-rated 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 hyper-Oscar E Hernández, Eduardo E Zurek, David Vera, Alfonso Cepeda-Emiliani polarizing 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:

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As for the IPSP rate (IPSP rate ), it can be estimated as follows:

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).Fuente: datos tabulados por los autores.This geometrical model was obtained from reference [44].The cell area using the procedure described in [52] G pass (S/cm 2 ) 5.0x10 -5 5.0x10 -5 5.0x10 -5 5.0x10 -5

RESULTS
The results of this work are focused, first, on the analysis of the relationships between f max in burst 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 h and the corresponding f max values (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/ cm 2 , 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 h = T h2 -T h1 is full width at half maximum.T fmax is the 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/ cm 2 , 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.The solid line in Figure 2B shows the relation between f max and T fmax and its points fitted to a growth quadratic equation (Eq. 1, with R 2 = 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 max increases by 1 kHz for each temperature increase of 1°C; and at T = 40°C, f max increases 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 max /dT increases by 2.485 x10 -3 Hz/°C for 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 R 2 = 0.99901), as shown by Figure 2C.1.0529 kHz represents the maximum value that f max 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/cm 2 .
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/cm 2 , df max /dG T decreases by 0.11 kHz/(1x10 -4 S/ cm 2 ) for each unit increase (1x10 -4 S/cm 2 ) in maximum Ca +2 conductance.For G T = 8x10 -4 S/cm 2 , f max decreases by 0.08 kHz/(1x10 -4 S/ cm 2 ) for each G T increase of 1x10 -4 S/cm 2 . ( 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 R 2 = 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.
) (5)   Figure 3B shows the relationship between ΔT h and f max .The points (ΔT h , f max ) were fitted to a growth exponential equation (eq.6, with R 2 = 0.96501).15.321°C represents the maximum value that ΔT h can reach independently of the f max value.
This saturation value occurs if f max tends to infinity.
) (6) From equations 5 and 6, it can be observed that f max = 0.086 kHz and G T = 6.37 S/cm 2 represent the values at which ΔT h reaches a value equal to 63% of its maximum possible value (15.321°C), i.e.ΔT h = 9.65°C.

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 The following observations highlight some of the most relevant results of this study and their electrophysiological implications.
Figure 2B and Equation 1show 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 max limit for any temperature increase, figure 2C and equation 3 show that for high G T values, maximum firing frequency of the TRNnm always tends to an f max limit.For the model used in this work, the limit tends to f max = 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

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 f max 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 h vs. f max ) was fitted to an exponential growth (see Equation 6).This equation shows that if specific TRNn display bursting activity with f max values, they can also display this type of behavior through a wide range of temperatures.
Finally, and similar to the above relationship (ΔT h vs. f max ), ΔT h vs. G T 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 G T .

Figure 1 .
Figure 1.Burst simulation in the neuronal soma after trains of random stimuli were applied on the distal dendrites.Burst simulation in the neuronal soma after trains of random stimuli were applied on the distal dendrites.A) Note the measuring point (black arrow) on the soma and the stimuli points (gray dots) on the distal dendrites.B) The train of random current stimuli with negative and positive amplitudes.C) Burst neuronal response due to random stimuli in the amplitude range IA[-0.25,0.1043] nA applied to the distal dendrites.Note that the burst in C has the same characteristics of the typical burst (reference burst) shown in [43, 15].

Figure 2 .
Figure 2. Frequency changes (f) during burst mode responses for different G T values.A) The variation of f for G T = 8x10 -4 S/cm 2 and for different temperatures was fitted to a polynomial equation.For

Figure 3 .
Figure 3. Relationship between the representative temperature range (∆T h ) at which the TRNnm responds in burst mode, the maximum calcium conductance (G T ), and the maximum burst frequency (f max ).In both cases, A) and B), the points of the curves are fitted to an exponential growth equation.

Table 1 .
Geometrical characteristics of the TRN model

Table 2 .
Active and passive electrical conduction properties of the neuronal model

Table 3 .
Formulas and constants using to model the variations of the intracellular concentration of Ca+2 res Fuente: datos tabulados por los autores.

Table 4 .
Formulas and constants used to model the activation processes of IT current

Table 5 .
Formulas and constants used to model the fast sodium channels

Table 6 .
Formulas and constants used to model the fast potassium channels Fuente: datos tabulados por los autores.
T .From Equations 5 and 6, it can also be inferred that if G T = 6.37 S/cm 2 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 ΔT h > 9.65°C.