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[MATH] (Koch, 1998). The average synaptic conductance is [MATH] , which in the physiologically valid limit [MATH] yields [MATH] . Neurophysiological data indicate (Markram et al, 1997) that the release probability [MATH] is frequency dependent. Here, we follow Dayan and Abbott (2001) and assume that synaptic depression... |
[MATH] , where [MATH] is the release probability at 0 Hz, [MATH] is the depression time constant, and [MATH] is the parameter controlling the degree of depression (for [MATH] lack of depression, for [MATH] maximal depression). Thus, the total average synaptic current is |
[MATH] where [MATH] is the number of synapses (or presynaptic neurons) per neuron. Fluctuations around the average [MATH] cause the neuron to fire an action potential with an average firing rate [MATH] Glutamate synapses with non-NMDA receptors are used more frequently for regular transmission (NMDA receptors are block... |
and K ions, which is the reason for the presence of [MATH] and [MATH] terms in the dynamics of [Na] and [K] in Eq. (1), where [MATH] |
The parameter [MATH] can be determined in a few steps. First, we can write the synaptic current [MATH] as [MATH] where [MATH] and [MATH] are K and Na |
conductances (voltage independent) at glutamatergic synapses. Since for [MATH] the synaptic current [MATH] , we have [MATH] Second, from the condition [MATH] we obtain [MATH] , which leads to |
[MATH] . Note that [MATH] is voltage dependent. Consequently, the synaptic contribution to changes in intracellular Na concentration is given by |
[MATH] The synaptic contribution to changes in intracellular K is [MATH] Equation for voltage dynamics in Eq. (1) assumes that the membrane is equipotential, i.e., axons and dendrites have on average equal potentials. This assumption neglects spatio-temporal effects associated with action potential propagation. The val... |
Adding appropriately sides in Eq. (1) we obtain the “charge conservation” equation: [EQUATION] This equation reflects the fact that non-zero membrane potential is caused by concentration gradients of charged ions across membrane, and temporal changes in potential are directly related to the temporal changes in ionic co... |
[MATH] . Thus, on the long-time scale, the right hand side of Eq. (2) is close to 0, which implies that [Na] + [K] [MATH] . That is, changes in [Na] directly determine changes in [K], and practically there is no need to solve the equation for the potassium dynamics. Values of neurophysiological parameters used in this ... |
2.2 Neuroanatomical relationships. In studying metabolic and thermodynamic properties of brain tissue the following neuroanatomical relations were used: (i) volume density of synapses [MATH] |
is brain size independent, where [MATH] is the number of neurons and [MATH] is the cortical gray matter volume (Braitenberg and Schuz, 1998; DeFelipe et al, 2002); (ii) the fraction |
[MATH] of the gray matter volume taken by fibers (axons and dendrites) is approximately constant and about 2/3 (Braitenberg and Schuz, 1998); (iii) white matter volume [MATH] scales with gray matter volume [MATH] as [MATH] (Zhang and Sejnowski, 2000), where [MATH] and [MATH] are expressed in cm |
The relationship between neuron’s surface area [MATH] and gray matter volume [MATH] can be determined as follows. The gray matter volume is |
[MATH] , where neuron’s volume [MATH] with [MATH] [MATH] denoting average (unmyelinated) axon and dendrite length in the gray matter and [MATH] [MATH] denoting their corresponding diameters (neuron’s soma is neglected as it is small). The parameter [MATH] is the fraction of volume taken by non-fibers (synapses, blood v... |
[EQUATION] which is proportional to the neuron’s volume to surface ratio, i.e., [MATH] . In fact, if volumes of axons and dendrites are equal, which seems to be neuroanatomically valid in the gray matter (Braitenberg and Schüz, 1998; Chklovskii et al, 2002), then [MATH] represents a harmonic mean of axonal [MATH] and d... |
[MATH] diameters, that is [MATH] . This formula can be derived by noting that axon length [MATH] (equal volumes of axons and dendrites), and substituting [MATH] to Eq. (3). Note that for constant [MATH] , the effective diameter [MATH] |
decreases with decreasing [MATH] . In the limit [MATH] , we have [MATH] For neuroanatomical values of [MATH] [MATH] m and [MATH] |
[MATH] in mouse (Braitenberg and Schüz, 1998), we obtain the effective fiber diameter [MATH] [MATH] m. Since the average values of cortical axon and dendrite diameters are roughly invariant with respect to brain size (i.e., they should be more or less the same for mouse and human; Braitenberg and Schüz, 1998), we can e... |
[MATH] m for illustrative purposes throughout the paper. Apart from that, occasionally we will also show results for the smallest experimentally known axons of diameter 0.1 [MATH] m corresponding to [MATH] |
[MATH] (for [MATH] [MATH] m), and for the smallest molecularly feasible axons (neurites) of diameter 0.06 [MATH] m (axons without Natrium channels that lack action potentials; Faisal et al, 2005), corresponding to |
[MATH] [MATH] m (also for [MATH] [MATH] m). With the definition of [MATH] we can relate the total surface area of neurons, [MATH] , to the gray matter volume [MATH] as |
[EQUATION] and relate the surface density of synapses along axons and dendrites, [MATH] , to synaptic density [MATH] as [EQUATION] |
These relations are used below to rewrite Eq. (1) in a more convenient form for numerical and analytical calculations. 2.3 Explicit voltage and ionic dynamics. |
With the modifications and interdependencies between parameters described in Subsections 2.1 and 2.2 we can rewrite Eq. (1) as: [EQUATION] |
where the release probability modulated by synaptic depression is [MATH] (Dayan and Abbott, 2001). These equations are solved both numerically (using the 4th-order Runge-Kutta method) and approximately analytically. Fast synaptic fluctuations denoted symbolically in the top equation of Eq. (6) cause the neuron to fire ... |
2.4 Electric power /K -ATP pumps. Since the metabolic rate in white matter is 3-4 times lower than in gray matter (Siesjo, 1978; Karbowski, 2007), the white matter contribution to the total brain metabolic power is neglected. The average power [MATH] dissipated by [MATH] neurons in the gray matter is given by |
[EQUATION] where the integral represents the electrical work performed by Na /K -ATP pumps during the long time [MATH] , much larger than the average interspike interval [MATH] . This work goes for removing 3 Na |
ions and importing 2 K ions against their electrochemical gradients, which cost 1 ATP molecule (Kandel et al, 1991). Opposite signs in front of |
[MATH] and [MATH] indicate the fact that Na and K ions move in opposite directions through the membrane. Integral in Eq. (7) can be estimated by noting that in the long-time limit the pump current [MATH] assumes its average value (with some fluctuations around, see the Results) given by |
[MATH] where [MATH] is the average long-term sodium concentration (determined in the Results section). In this limit, the average value of voltage [MATH] is approximately equal to its resting value [MATH] . Thus, the electric power [MATH] is |
[EQUATION] where [MATH] and [MATH] , with the extracellular Na and K concentrations (Hille, 2001): [MATH] mM, and [MATH] mM. 2.5 Mechanisms of brain cooling. |
The major contribution to the cerebral metabolic rate or heat constitute Na /K -ATPase (Astrup et al, 1981; Erecinska and Silver, 1989; Rolfe and Brown, 1997; Ames, 2000). Therefore, in this paper, the heat coming from other reactions such as glycolysis (conversion of glucose to ATP) is assumed to be less important and... |
pumps is transfered by conduction and circulating cerebral blood flow, whereas at the surface (scalp) the heat is removed mostly by convection/conduction and radiation (Fig. 1). Heat conduction (convection) is associated with the existence of a temperature gradient between the brain or scalp and the external environmen... |
The equation governing heat balance and spatial distribution of brain temperature [MATH] has the form (Nelson and Nunneley, 1998; van Leeuwen et al, 2000; Sukstanskii and Yablonskiy, 2006): |
[EQUATION] where [MATH] is the arithmetic mean of the thermal conductance of brain tissue (including cerebrospinal fluid, skull, and scalp at the edge), |
[MATH] and [MATH] denote blood’s density and specific heat, CBF is the cerebral blood flow rate expressed in sec -1 [MATH] is the incoming blood (arterial) temperature equivalent to body core temperature, and [MATH] is the heat (per time unit) generated in the gray matter due to Na /K -ATPase. In general, [MATH] |
because [MATH] is the useful work (per time unit) performed due to hydrolysis of ATP and release of [MATH] of free energy (per time unit). In Sec. 3.5 we estimate the relative magnitude of [MATH] and |
[MATH] , i.e., we calculate the efficiency of the sodium pump. The parameter [MATH] denotes brain volume, i.e., [MATH] (brain geometry is modelled as half of a ball; Fig. 1). The parameters |
[MATH] and [MATH] denote brain tissue density and specific heat and they are approximately equal to [MATH] and [MATH] , respectively. Only steady-state regime of Eq. (9), i.e. [MATH] is considered below. |
The boundary condition imposed on Eq. (9) has the following form: [EQUATION] where [MATH] is the brain’s radius given by [MATH] [MATH] is the Stefan-Boltzmann constant, [MATH] is the heat convection/conduction coefficient between the scalp and the outside environment, and [MATH] and [MATH] denote the scalp and environm... |
The heat removed from the brain through the conduction is given by [MATH] . The heat removed by the circulating cerebral blood is given by [MATH] Therefore, by integrating Eq. (9) for the whole brain volume, at the steady-state, we obtain: |
[EQUATION] where the conduction term [MATH] is a sum of scalp convection/conduction and scalp radiation, i.e., [MATH] , with [MATH] and |
[MATH] Values of all thermodynamic parameters are presented in Table 1. Results 3.1 Sodium influx during a single action potential. |
When synaptic fluctuations cause the neuron to fire an action potential, the Na concentration first rise and then slowly decays to its equilibrium value, due to the workings of the Na /K pump. Sodium influx during a single action potential can be estimated using Eqs. (2) and (6). It is composed of the two contributions... |
[MATH] . The total influx is given by (see Appendix A): [EQUATION] where [MATH] denotes “correction” to the effective capacitance coming from the prolonged Na channels activation and is given by |
[MATH] . Value of [MATH] is taken from numerical simulations of Eq. (6), and is approximately [MATH] msec. This number enables us to determine the parameter [MATH] in Eq. (12), which we find to be [MATH] |
[MATH] F/cm for typical resting values of voltages: [MATH] mV, [MATH] mV (for [Na]= 12 mM, [K]= 155 mM; Hille, 2001) and [MATH] mV. It should be remembered, however, that [MATH] is not a constant, but it varies slightly depending on the level of intracellular sodium concentration [Na] via [MATH] and [MATH] (see Fig. 2B... |
The immediate conclusion from Eq. (12) is that the amplitude of the sodium influx increases with decreasing the effective fiber diameter [MATH] and this agrees with a direct numerical integration of Eq. (6), as is shown in Fig. 2C. Based on the above values, we find from Eq. (12) that for [MATH] |
[MATH] m (harmonic mean of average axon and dendrites diameters in mouse; Sec. 2.2) the sodium influx [MATH] mM, which corresponds to |
[MATH] Na ions per [MATH] For a mouse neuron with equal volume of axons and dendrites, and with [MATH] cm and [MATH] [MATH] m, this gives the total influx of [MATH] Na ions. Direct numerical integration of Eq. (6) for [MATH] |
[MATH] m yields a similar sodium influx, [MATH] mM, which corresponds to the total influx of [MATH] Na ions for a mouse neuron. The influx of [MATH] Na ions during an isolated action potential obtained above is comparable to the estimate of Attwell and Laughlin (2001), who used a different, phenomenological, approach a... |
[MATH] . The total charge influx [MATH] in Attwell and Laughlin (2001) is [MATH] if we neglect a small soma contribution. The prefactor of 4 was chosen by these authors to a large extent arbitrary, and it comes from the effect of simultaneous activation of Na and K channels, which is analogous to the presence of the [M... |
is the sodium influx in the Attwell and Laughlin (2001) formulation, we obtain [MATH] , where [MATH] is the effective fiber diameter defined in Eq. (3). The formula for [MATH] is very similar to the formula (12), except for the numerical factor in front, which is 13.6 in Eq. (12). This results from the fact that numeri... |
[MATH] for low firing rates or intracellular sodium concentrations (Fig. 2B). It is also interesting to note that the Attwell-Laughlin (2001) formula does not account for changes in the intracellular Na concentration due to repetitive firing, as opposed to Eq. (12) that includes such changes through [MATH] . In this se... |
3.2 Sodium build-up due to repetitive firing. When the neuron fires repeatedly, its intracellular sodium accumulates with every spike because Na /K pump is slow and cannot remove all Na promptly (Fig. 3A,B). Since every spike introduces |
[MATH] of sodium electric charge, and this process is fast, we can approximate Eq. (6) for the Na dynamics as [EQUATION] where [MATH] is the sodium current associated with sodium channels and synaptic contribution at rest, and it is given by |
[MATH] . The delta functions present in Eq. (13) represent spikes of Na influx at times [MATH] Simulation of Eq. (13) is shown in Fig. 3C, and it resembles the simulation of the original Eq. (6) (see Fig. 3A). |
We can further simplify Eq. (13) at the long-time limit, in which we substitute for the right hand side of Eq. (13) its temporal average. In particular, the temporal average of [MATH] |
is equal to the firing rate [MATH] , and thus [EQUATION] where [MATH] is the temporal average of [Na]. A comparison of the time dependence of [MATH] from Eq. (14) with the time dependence of [Na] coming from Eq. (13) is presented in Fig. 3C. The equilibrium value of [MATH] , denoted as [Na] av |
is determined from the condition [MATH] , with the help of Eq. (12). [Na] av satisfies the following equation: [EQUATION] Note that the right hand side of Eq. (15) also depends on [Na] av |
through [MATH] and [MATH] . Thus, we can find [Na] av only numerically, either from Eq. (15) or from a direct simulation of Eq. (14). |
In Fig. 4, we compare the dependence of [Na] av on firing rate that comes from Eq. (15) with the dependence that comes from a direct numerical integration of Eq. (6). Overall, the formula (15) provides a relatively good approximation to the numerical solution, especially for low firing rates (Fig. 4). For this reason, ... |
It is interesting to note that the average [Na] av is weakly dependent on fiber diameter, especially for small firing rates (Fig. 4). The reason for this is that only the current [MATH] in Eq. (15) contains synaptic contribution with [MATH] , and for low frequencies |
[MATH] is small. It should be kept in mind, however, that sodium fluctuations around [Na] av do grow with decrease in fiber diameter (Fig. 3B), because for thin fibers the amplitude of sodium influx is large [MATH] ) and the relaxation time constant is short ( [MATH] ; see Appendix B). Enhanced intracellular sodium flu... |
[MATH] m corresponding to the smallest physically possible axons of [MATH] [MATH] m (Faisal et al, 2005) the intracellular sodium concentration can occasionally peak to extracellular levels (145 mM; Hille 2001) just for 40 Hz of repetitive firing. For [MATH] |
[MATH] m (corresponding to [MATH] [MATH] m) this takes place for [MATH] Hz. These considerations suggest that very thin fibers are not beneficial for neuron’s electrical properties. |
3.3 ATP utilization rate of Na /K pump, glucose metabolism, and firing rate in mammals. For pumping out Na and pumping in K , the Na /K |
pump uses ATP molecules. The average ATP utilization rate of a single neuron per its surface area [MATH] is equal to [MATH] Generally, it increases with [Na] av and thus with the firing rate [MATH] in a non-linear fashion (Fig. 5A). For low frequencies [MATH] there exist a linear regime, while for high [MATH] , the ATP... |
[MATH] (Fig. 5A). Typical values of ATP utilization for frequencies in the range [MATH] Hz are [MATH] ATP molecules per [MATH] per second. In the linear regime, the increase of firing rate by 1 Hz leads to the increase of the ATP rate by about [MATH] |
[MATH] mol/(cm sec). It is important to point out that in previous phenomenological models of ATP utilization rate (Attwell and Laughlin 2001; Lennie, 2003) only linear regime was assumed in estimations. Thus, the present more detailed model extends these calculations into the non-linear regime. |
The average firing rate in mammalian brains can be determined indirectly from the cerebral glucose utilization rates CMR glu (expressed in mol/(cm [MATH] s) ). If we assume that the ATP activity of the neural pumps constitutes the major contribution to the gray matter metabolism (Astrup et al, 1981; Erecinska and Silve... |
[MATH] , from which we obtain, using Eq. (4), that [EQUATION] We can write CMR glu in an equivalent form, which contains neurophysiological parameters explicitly. Using Eq. (15), we obtain: |
[EQUATION] Dependence of CMR glu on firing rate [MATH] is plotted in Fig. 5B, and it is practically the same as the dependence of [MATH] on |
[MATH] , as the two quantities CMR glu and [MATH] are proportional. The non-linear part of the dependence comes from the fact that [MATH] |
and [MATH] decrease for high frequency (via [Na] av ). The CMR glu vs. [MATH] relationship (Eq. 17) enables us to find average frequencies for several mammalian species for which empirical values of CMR glu are known (Fig. 5C). In general, the average firing rates are rather low, from [MATH] 1.7 Hz for human to [MATH] ... |
3.4 Biphasic dependence of pump power on frequency. The electric power /K pump in a single neuron (per surface area) is determined in two ways. First, from a direct numerical integration of Eq. (7) with time dependent voltage [MATH] and pump current |
[MATH] . Second, from the approximate analytical formula (8) with the help of derived Eq. (15). Both methods yield similar results (Fig. 6A), which indicates that the approximation (8) is reliable, especially for low firing rates. We can rewrite Eq. (8) for the total electric power generated in the gray matter in a mor... |
[EQUATION] Alternatively, we can use Eq. (15) to relate the pump current to the neurophysiological parameters. In this way, we obtain: |
[EQUATION] Note that synaptic depression via [MATH] reduces the power [MATH] In Eq. (19) the first term in the large bracket represents a very small sodium influx at rest, the second term corresponds to the background dendritic synaptic activity, and the last term comes from Na influx due to action potentials. The rela... |
of the total power. Fig. 6A indicates that [MATH] depends biphasically on firing rate [MATH] . This is a non-intuitive result, following from the fact that |
[MATH] is a product of two terms: [MATH] and [MATH] The first of them increases monotonically with [MATH] , whereas the second decreases with [MATH] because [MATH] and [MATH] depend on frequency via [Na] av . This biphasic dependence of [MATH] on frequency has interesting implications for thermal properties of brain ti... |
The sodium pump power [MATH] also depends inversely on the fiber diameter [MATH] (Eq. (19) and Fig. 6B), and proportionally on the gray matter volume [MATH] . Thus, too thin fibers are metabolically expensive. |
The power directly to the glucose cerebral metabolic rate CMR glu if we combine Eqs. (16) and (18). The resulting relationship is: |
[EQUATION] Thus, the power generated scales linearly with the glucose consumption rate. However, it should be kept in mind that the proportionality factor is not a constant, but changes with firing rate. |
3.5 Efficiency of the Na /K pump. Let us estimate the efficiency of the sodium pump, i.e., how much energy does it use for pumping out 3 Na and pumping in 2 K |
ions given an available energy from ATP hydrolysis. Energy from hydrolysis of 1 ATP molecule goes for performing the useful work of |
[MATH] , or equivalently 1 mole of ATP performs the work of [MATH] , where [MATH] is the electron charge. On the other hand, hydrolysis of 1 mole of ATP generates [MATH] of free energy. Thus, the efficiency of the process is given by |
[MATH] . The value of [MATH] depends to some extent on the internal chemical (ionic) state of the cell, and it has been reported to be in the range from 48 kJ/mol (Jansen et al, 2003) to 62 kJ/mol (Erecinska and Silver, 1989). This leads to the pump efficiency of [MATH] |
[MATH] (for typical resting values of voltages: [MATH] V, [MATH] V, [MATH] V; Hille (2001)). Because of the high efficiency of the Na /K -ATPase, in what follows, we make an approximation in which we equate the heat released in the gray matter due to hydrolysis of ATP ( [MATH] in Eq. 9) with the electrical power dissip... |
We can estimate the heat rate for the gray matter of human brain from Eq. (20). Taking [MATH] cm (Stephan et al, 1981), [MATH] mol/(cm [MATH] sec) (or 0.34 [MATH] mol/(cm [MATH] min); e.g. Clarke and Sokoloff, 1994), and for the above values of voltages we obtain [MATH] |
Watts. Given a possible increase of this heat value by up to 27 [MATH] due to pump efficiency, this result does not differ much from other estimates of heat in gray matter. For example, Aiello and Wheeler (1995) used mass specific heat generation of 11.2 W/kg (based on older experimental data), which yields 7.8 Watts f... |
3.6 Scaling of cerebral blood flow with brain size. Cerebral blood flow CBF is important in controlling brain temperature (see the next subsection). The dependence of physiologically averaged CBF on brain volume can be found from the empirical data available in the literature. The results in Fig. 7 for 6 mammals spanni... |
[MATH] sec -1 for human. 3.7 Brain temperature vs. frequency and fiber diameter, and efficiency of brain cooling The spatial distribution of brain temperature [MATH] is found by solving Eq. (9) with [MATH] , and the result is given by (see Appendix C): |
[EQUATION] where [MATH] . The parameter [MATH] characterizes the inverse of the length of “transition” region from the scalp to the brain’s interior where temperature is inhomogeneous (Fig. 8A). For human [MATH] cm, and it is much smaller than human brain radius ( [MATH] cm), and thus heterogeneity is present only at t... |
cm, i.e., it is comparable with the mouse brain radius ( [MATH] cm), implying that in very small brains cerebral temperature is inhomogeneous in the whole volume (Fig. 8A). The scalp temperature [MATH] present in Eq. (21) is determined self-consistently (see Appendix C) from the condition [MATH] , and it decreases very... |
Deep brain temperature [MATH] is slightly larger than the blood temperature by [MATH] C for all analyzed mammals, except mouse (Table 2). This value lies in the range of values observed experimentally (Hayward and Baker, 1968; Nybo et al, 2002; Kiyatkin, 2007). In general, however, [MATH] is very weakly species specifi... |
[MATH] which shows that the boundary temperature (and scalp cooling) becomes unimportant in this limit. From this formula it follows that the deep brain temperature is greater than the blood (or core body) temperature by the quantity proportional to the ratio of CMR glu to CBF, in agreement with Yablonskiy et al (2000)... |
Deep brain temperature [MATH] depends significantly on the effective fiber diameter [MATH] (Fig. 8B,C). For very small [MATH] (corresponding to very small [MATH] ), the temperature [MATH] tends to diverge, which is a direct consequence of the fact that [MATH] also diverges for [MATH] (or equivalently [MATH] ). In this ... |
In contrast to the monotonic dependency of [MATH] on [MATH] , its dependence on firing rate is biphasic (Fig. 8D). The origin of this non-monotonic behavior is the dependence of [MATH] on [MATH] , which is also biphasic (see Fig. 6A). This result has surprising thermodynamic consequences, namely too high levels of neur... |
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