Question

Universidad ue Europea Action potential in neurons Dendrites Neurons (nerve cells) are the core components of the nervous system. They provide critical functions such as control ling muscle activity, sensing thinking, remembering, and regulating glandular secretions. Soma Electrical signals and chemical messengers are used to transmit and receive information between them. The neurons consist of: Nclen Asos The cell body (soma) and its processes Dendrites: the branched projections of a neuron that act to propagate the electrochemical stimulation received from other neural cells to the cell body The axon: a nerve fibre that conducts electrical impulses away from the cell body to signal other neurons As a nerve impulse travels down the axon, the change in voltage v of a neuron with respect to time follows the differential equation dv =1.0+ dt e01t 1+0.02t over the course of 100 ms, where t is measured in milliseconds and v(0)-70 mV in millivolts. We start at (a) Integrate the previous differential equation to express voltage as a function of time, v(t), and sketch its graph for t E (0,100 (b) What is the voltage after 100 ms? A neuron in your brain sends a charge down an 80 cm long axon toward your hand at a speed of 10 m per second. At the time when the charge reaches your elbow, the voltage in the axon is -70 mV except on the 6 cm long piece between 47 and 53 cm from your brain. On this piece, the voltage is v(x)-70.0+10.0[9.0-(x-50.0)2 where v(x) is the voltage (in mV) at a distance of x centimeters from the brain. (c) How long will it take the information to get to your hand? How long did it take to reach your elbow? (d) Sketch a graph of the voltage along the whole axon.

Answer 1

Given, dv =1+ 20.01 dt 1+0.02 This is a variable separable differential equation 1 dv 1+ e t 1+0.02t Integrate both sides. 1 -In1+0.02 0.02 v=t+ 0.01 v = t+50ln1 0.02r-100e c Now, v(0)-70 So, +50 ln1|-100e +c c 100 70 = 30 So, v t+50lnl+0.02-100e0 +30 .01n The graph of the solution is shown below. TEXAS INSTRUMENTS TI-83 Plus y1=X+50*1n (13 0.02*%)- 10.- Y-50.21477