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In the superficial regions brain temperature is always smaller than the blood temperature (Fig. 8A; Table 2). This fact has important consequences for the heat transfer in the mammalian brains. In deep brain regions cerebral blood flow plays the role of a coolant, whereas in the superficial regions it serves as a stron... |
(Table 2). In general, the cooling mechanisms discussed in subsection (2.5) depend on brain temperature distribution, on scalp temperature [MATH] , and on brain volume [MATH] . Therefore, the efficiency of cerebral cooling is species specific. In particular, the relative importance of the two major mechanisms of heat t... |
is comparable to the heat warming rate via blood flow [MATH] (Table 2). The reason for such a strong warming through the cerebral blood is that it must compensate heat loss due to scalp convection/conduction |
[MATH] and radiation [MATH] , which are much larger than the metabolic rate [MATH] . On the other hand, for large brains, the cooling rate [MATH] is almost twice as large as the warming rate |
[MATH] , because heat production due to metabolic activity [MATH] is more significant than for small brains (Table 2). On the scalp, the heat transfer rate is dominated by convection/conduction [MATH] as it is twice the radiation rate [MATH] |
3.8 Thermal bounds on fiber diameter and length. Mammals are able to sustain brain temperatures up to about 42 without causing brain damage (Gordon, 1993; Kiyatkin, 2007). Above these temperatures molecular changes in neurons and synapses become critical and irreversible. In what follows, we estimate the thermal bounds... |
[EQUATION] where we used Eq. (19) for [MATH] . The minimal effective fiber diameter [MATH] depends very weakly on gray matter volume [MATH] (Fig. 9A). This suggests that brain size is not a critical factor determining thermodynamic safety of the cerebral tissue. In other words, thermal properties of mouse and elephant ... |
The minimal effective fiber diameter [MATH] depends stronger and biphasically on firing rate [MATH] (via [MATH] [MATH] , and [Na] av (Fig. 9B). For very low firing rates ( [MATH] ), the expression (22) simplifies, and we obtain in this limit: |
[EQUATION] which yields [MATH] nm (values of [MATH] and [MATH] are for human brain). For intermediate values of [MATH] , for which [MATH] has a maximum, the value of [MATH] can be in the range |
[MATH] [MATH] m (Fig. 9B), which corresponds to a bound on axon diameter [MATH] [MATH] m. The latter value is only about 5 times larger than the membrane thickness (Koch, 1998). The term proportional to |
[MATH] in the denominator of Eq. (22) is the background synaptic contribution with frequency dependent depression inside [MATH] . Due to this depression the synaptic contribution is bounded from above and consequently does not diverge as a function of [MATH] . Thus, synaptic depression not only reduces the power [MATH]... |
Average value of the empirical fiber diameter [MATH] in the gray matter is brain size independent and is about [MATH] [MATH] m (harmonic mean of average axon and dendrite diameters in mouse, as defined in Sec. 2.2, Eq. (3)). This value is about 10 times larger than the largest value of |
[MATH] . Because of this, mammalian brains operating under normal physiological conditions are rather safe from excessive overheating that would cause brain damage, for all ranges of frequency. The situation could be more tricky in hot environments in which |
[MATH] were severely reduced, due to increase in body temperature and hence cerebral blood temperature. The margins of thermal safety, could be also compromised in pathological conditions. For example, abnormalities associated with strongly reduced cerebral blood flow CBF or compromised synaptic depression, could signi... |
The thermal lower bound on fiber diameter also determines the upper bound on fiber length per neuron [MATH] , as is evident from Eq. (4). Taking the neuron’s surface area [MATH] , we get |
[MATH] , where [MATH] [MATH] m. For human brain with [MATH] cm (Stephan et al, 1981) and [MATH] (Haug, 1987; Braendgaard et al, 1990) this yields [MATH] meters. For mouse brain with [MATH] cm |
and [MATH] (Braitenberg and Schuz, 1998) we get [MATH] meters. Neuroanatomical data for mouse gray matter indicate that the average fiber length per neuron is [MATH] cm (Braitenberg and Schuz, 1998), which is 2-3 orders of magnitude below its upper thermal bound. A similar conclusion holds for the human brain. |
Taken together all these results suggest that thermodynamics does not restrict neuroanatomical parameters in any dramatic way, because they are far away from their thermal bounds. |
Discussion This article investigates thermodynamic properties of brain tissue and corresponding physical limits on neural anatomy caused by heat balance in the brain. It is found that, in general, the lower and upper limits on fiber dimensions are unattainable for normal values of physiological parameters such as cereb... |
The conclusions of this paper are based on calculating neural metabolic power and spatial distribution of heat dissipated. Since majority of metabolic energy in neurons goes to pumping out sodium ions (Ames, 2000; Astrup et al, 1981; Erecinska and Silver, 1989; Rolfe and Brown, 1997), the first step in determining neur... |
The issue of brain cooling is not a classic problem of volume to surface ratio, as it is the case with natural, non-designed, physical objects. Instead, the brain cooling could be compared to the cooling of combustion heat engine, which receives a liquid coolant. In the brain the role of the coolant is played by the ce... |
The interesting result is that the power depends biphasically on neural frequency of firing, and inversely on the effective fiber diameter (Fig. 6). As a consequence of this, brain temperature |
[MATH] can depend biphasically on frequency as well, if cerebral blood flow does not change with frequency (Fig. 8D). Thus [MATH] increases with frequency but only up to a certain point above which it slightly decreases with further increase in frequency. The increase of brain temperature in response to activation (hig... |
The negative correlation between the power generated in the brain and the effective wire diameter (Fig. 6B) suggests that thin fibers are energy expensive. The effective fiber diameter [MATH] is defined as a harmonic mean of axon [MATH] and dendrite [MATH] diameters (Sec. 2.2), with the assumption that volumes of axons... |
Another negative consequence of having too thin fibers is that cerebral tissue temperature increases inversely with fiber diameter (Fig. 8C), i.e., [MATH] . Thus, brain regions reach in very thin wire can heat up excessively. As an example, the temperature of the thinnest known axons with diameter 0.1 [MATH] m (Faisal ... |
[MATH] [MATH] m), relative to blood temperature, should be about 2.5 times larger than the corresponding relative temperature of 0.3 [MATH] m axons (mouse cerebral cortex; Braitenberg and Schuz, 1998; corresponding to |
[MATH] [MATH] m), assuming both axons have the same firing rate. Thus, if typical relative temperatures above blood temperature are [MATH] |
for 0.3 [MATH] m axons (Hayward and Baker, 1968), then for 0.1 [MATH] m axons the corresponding relative temperatures would be [MATH] |
C. This result suggests an explanation why in the peripheral nervous system sensory fibers responsible for high-threshold heat sensation (so-called C-fibers) are much thinner ( [MATH] |
[MATH] m; Kandel et al, 1991) than other sensory fibers (A-type or B-type with diameters [MATH] [MATH] m). It might be that they warm up easier, although these fibers also serve other functions (Craig, 2003). |
This study finds that the lower thermal bound on the effective fiber diameter [MATH] is strongly frequency dependent, but it is finite due to synaptic depression. Values of [MATH] are in the range [MATH] nm (or [MATH] |
[MATH] m; see Fig. 9B). These values can be translated to the corresponding thermal bounds on axon diameter [MATH] , which are in the range [MATH] |
[MATH] m (assuming that the dendrite diameter [MATH] [MATH] m and it does not change). The average value of cortical axon diameter is [MATH] |
[MATH] m, and thus it is 12-1500 times larger than these limits, therefore, on average, mammalian brains operate in the safe thermal zone. However, it should be also kept in mind that axon diameter displays some variability, and the thinnest axons can reach 0.1 [MATH] (Faisal et al, 2005). This value is still about 4 t... |
[MATH] m) is smaller than corresponding bounds imposed by structural constraints and noise, which are respectively [MATH] [MATH] m and [MATH] |
[MATH] m (Faisal et al, 2005). The theoretical bound on axon diameter implies a corresponding lower bound on the speed of signal propagation in the gray matter. Let us estimate the upper limit on temporal delays. For unmyelinated fibers (prevalent in gray matter) velocity of signal propagation is proportional to the sq... |
[MATH] m the propagation velocity is 2.3 mm/msec (Koch, 1998), which implies that for axons with the boundary thickness of [MATH] |
[MATH] m, we have velocities 0.37 mm/msec. This gives for the maximal known extent of axons in the gray matter of 9-10 mm (for macaque monkey visual cortex; Amir et al, (1993)) the upper limit on delays in the range 24-27 msec. Thus, apparently, thermodynamics of heat balance in the gray matter does not tolerate tempor... |
The problem of finding bounds on fiber dimensions is similar in spirit to the approaches of “wire minimization” in the brain (Cherniak 1995; Murre and Sturdy 1995; Karbowski 2001, 2003; Chklovskii et al, 2002). It is hypothesized that this principle governs the organization of the mammalian nervous system at different ... |
The estimated firing rates in mammals are in the range from 1.7 Hz for human to 6.2 Hz for mouse, and they scale systematically with brain size, with the exponent [MATH] (Fig. 5C). The estimate for rat (5 Hz) is very close to that assumed by Attwell and Laughlin (2001), i.e. 4 Hz, which was based on weighted average of... |
The estimate of the heat generated in the gray matter may have some margins of error. First, the assumption of the equipotential neuron is only an approximation, because it does not include explicitly the spatio-temporal effects associated with action potential propagation and back-propagation. However, it is estimated... |
[MATH] [MATH] ; Sec. 3.5), the actual heat dissipated in the gray matter may be larger than [MATH] by [MATH] [MATH] . Therefore the total error from these two effects on the cerebral heat can theoretically reach 43 [MATH] We should also remember that some heat coming from the glucose to ATP conversion was neglected (i.... |
efficient). However, the comparison of the heat estimates for the human gray matter indicates that in fact the total heat error cannot be larger that 30 [MATH] (Sec. 3.5). There are also other sources of error that affect the temperature distribution in the brain, such as the neglect of cooling by scalp perspiration, a... |
The formulas (Eqs. 17-21) in this paper for the CMR glu , power dissipated in the gray matter, and brain temperature, may have practical use. These formulas as well as their possible future extensions can be used for assessing neural activity (firing rates) based on changes in temperature, CMR glu , and CBF. Moreover, ... |
Acknowledgments The work was partly supported by the Caltech Center for Biological Circuit Design. I acknowledge useful suggestions of the two anonymous reviewers. |
Appendix A: Sodium influx during an action potential. The duration of a typical action potential can be divided into two phases (Fig. 2A). During the first phase Na conductance [MATH] |
rises almost instantenously to its maximal value [MATH] and voltage [MATH] increases to its peak value [MATH] . The second phase is characterized by decline in values of [MATH] (to zero) and [MATH] |
(to values [MATH] ). The total Na influx during an action potential is [MATH] where superscripts (1) and (2) refer to the first and second phase, respectively. |
During the first phase, the intracellular potassium concentration practically does not change, because K channels are activated with a delay. Thus, by straightforward integration of Eq. (2), we obtain the Na influx during this phase as: |
[EQUATION] where the contribution proportional to [MATH] was neglected, since it is much smaller. Sodium influx during the second phase can be computed from the sodium dynamics of Eq. (6). Most of the time during this phase, the term proportional to [MATH] is much larger than the remaining two terms in Eq. (6). Thus, w... |
[EQUATION] where [MATH] is the duration of the second phase and [MATH] msec (from simulations). To simplify calculations, it is assumed that Na conductance and voltage depend on time in the following way: [MATH] |
and [MATH] These forms assure that for [MATH] , we have [MATH] and [MATH] , and for [MATH] we have [MATH] and [MATH] The latter value comes from simulations. After performing integral in Eq. (25) we obtain: |
[EQUATION] The fits in Fig. 4 are made for [MATH] . The total Na influx during an action potential is [MATH] and is given by Eq. (12) in the main text. |
It is also interesting to check the magnitude of correction coming from the fact that real neurons are not equipotential and action potentials propagate and back-propagate with a finite velocity. Let [MATH] be the spatial constant characterizing membrane potential homogeneity, and |
[MATH] the velocity of action potential propagation (along axon) and back-propagation (along dendrites). A simple way to account for the spatio-temporal dependence of Na conductance and voltage in Eq. (25) is to make the following rescalings: |
[MATH] and [MATH] where the factor [MATH] denotes a traveling wave of excitation along axon or dendrite with the spatial spread [MATH] , and the function |
[MATH] is the standard Heaviside function equal to 1 for [MATH] and equal to 0 for [MATH] . These rescalings assure that points along axon and dendrite separated by [MATH] from the soma (action potential initiation zone) receive an excitation after the delay time [MATH] . With these modifications, after some algebra, t... |
[EQUATION] where [MATH] is given by [MATH] The term proportional to [MATH] represents the first order correction and it is clear that the equipotential approximation corresponds to the case when [MATH] or [MATH] For physiological values of parameters: |
[MATH] mV, [MATH] mV, [MATH] mm/msec, [MATH] mm (Koch, 1998), with [MATH] msec and [MATH] , we obtain [MATH] . This implies that the equipotential assumption slightly underestimates the sodium influx during the second phase of sodium activation. Overall, this correction to the total Na influx [MATH] is small, [MATH] fo... |
Appendix B: Relaxation time constant for Na dynamics. In this Appendix the relaxation process of voltage and sodium to their equilibrium values is analyzed following a single action potential for a neuron that prior to that has been at rest for a long time. Computations below are performed in a very late phase of an ac... |
returned essentially to their resting values (or are close to them, i.e. [MATH] ). During this phase we can use a linear approximation on [MATH] and [Na] in Eq. (6). That is, we can expand [MATH] and [MATH] around their resting values |
[MATH] and [MATH] as: [MATH] [MATH] where [MATH] and neglect higher order terms. In this approximation the pump current [MATH] can be written as [MATH] , where |
[MATH] , and [MATH] The corresponding changes in [MATH] and [MATH] are [MATH] and [MATH] Next, using the facts that [MATH] and [MATH] we can solve Eq. (6) analytically. The equations governing [MATH] |
and [MATH] relaxations are given by: [EQUATION] The term on the right hand side of [MATH] dynamics that is proportional to [MATH] is much smaller than [MATH] and thus it can be neglected. This implies that changes in [MATH] and |
[MATH] due to sodium influx for an isolated action potential are very small. This leads to a simple exponential decay of [MATH] as |
[MATH] with the time constant [MATH] , where [MATH] is Na influx during the action potential given by Eq. (12). Thus, relaxation time constant is short for thin fibers, and it increases proportionally with fiber diameter. As an example, for [MATH] |
[MATH] m we obtain [MATH] sec (for [MATH] and [MATH] ), which is of the right order of magnitude (Abercrombie and Weer, 1978; Nakao and Gadsby, 1989). For [MATH] |
[MATH] m the time constant [MATH] is about 2 sec. Appendix C: Solution of the thermal balance equation. The steady-state limit of Eq. (9) can be rewritten in the form: |
[EQUATION] where [MATH] In computations, we assume [MATH] because of the high efficiency of Na /K -ATPase (Sec. 3.5). We look for the solution of Eq. (29) in the form: |
[MATH] Substituting this form in Eq. (29), we obtain [MATH] and we take only the positive solution as the one corresponding to the physical situation. From the boundary condition in Eq. (10) we obtain the coefficient |
[MATH] . Combining these results we find Eq. (21) in the main text. The scalp temperature [MATH] is determined numerically from the condition [MATH] , for all considered species. |
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