Matematyczne modelowanie mzgu (czyli o termodynamice) Jan - - PowerPoint PPT Presentation

Matematyczne modelowanie mózgu

(czyli o termodynamice) Jan Karbowski

University of Warsaw

Brain sizes

mouse rat human

Global brain metabolic scaling

Brains are energy expensive: brain scaling exponent (slope) = 0.86 > 3/4 for whole body metabolism (Karbowski, (Karbowski, BMC Biol 2007) BMC Biol 2007).

slope = 0.86 slope = 0.86

Non-uniform brain activity pattern

Neural activation

Information transfer ≈ ion flows

Voltage and Na-K dynamics

CS dV/dt = - gNaS(V-VNa) - gKS(V-VK) - gLS(V-VL)

• Ip - Is

UF d[Na]/dt = - gNaS(V-VNa) - 3Ip - αIs UF d[K]/dt = - gKS(V-VK) + 2Ip - βIs Ip = AS [Na]³/([Na]³ + θ³) gNa = gNa,max m³ h, gK = gK,max n

F – Faraday constant, U – neuron volume

Power generated by Na/K- ATP pumps PATP = (N/T) ∫dt (- 3Ip(V-VNa) + 2Ip(V-VK)) N – number of neurons in gray matter PATP = NSA[Na]a³ (3VNa - 2VK - Vr)/([Na]a³ + θ³)

Power generated – result (Karbowski – J. Comput. Neurosci. 2009)

PATP ~ (Ug/d) (gNa,o + syn.cond. + f(C+δC))

For thin fibers (d→0) the power PATP diverges and much heat is generated! Danger of brain overheating!

ATP, firing rate, and brain size

Heat removal from the brain

• brain tissue conductance,
• scalp conductance and radiation,
• cerebral blood flow (deep in the brain)

Heat balance equation

ρbr cbr ∂T/∂t = κ ∂²T/∂r² - ρbl cbl CBF(T-Tbl) + PATP/Ubr T – brain tissue temperature κ – brain thermal conductance Tbl – cerebral blood temperature r – spatial position usually Tbl < T and thus blood flow removes heat

Dependence of brain temperature on firing rate and fiber diameter

(mouse) (human )

Conclusions

• Brains use more energy than other tissues in the body.
• Thermal properties of mammalian brains depend

strongly on firing rates and axon diameter.

• Brains are safe from overheating (too thick axons).