HW5_Soln

HW5_Soln - BENG 1863 Winter 2010 Homework 5 Solutions...

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Unformatted text preview: BENG 1863 Winter 2010 Homework 5 Solutions Problem 1 [15 points]: A cell of spherical geometry with diameter 120 pm is bathing in a solution of 40 g NaCl and 1 g KCI in 10 L of deionized water at room temperature 20 °C. The cell has equilibrium potentials ENa = 80 mV and EK = -100 mV for Na+ and K+ respectively. a. Find the corresponding inside concentrations of Na+ and l<+ at equilibrium, and find the cell resting potential assuming equal permeabilities for Na+ and K, and zero permeabilities for other ion types. b. If the ion pumps that are maintaining the equilibrium potent als are now deactivated, find the resulting inside concentrations of Na+ and K+ at the new equilibrium. c. If the ion pumps are then reactivated to supply the original equilibrium potentials, estimate the total numbers of Na+ and K* ions that need to be pumped through the membrane. Indicate the direction of the Na+ and K+ ion transport. <1. 0mm WWW; Ama- No»; 23 D/Wt , my? [1‘03 im lOL z) LNWEO : :o-Oé I (OL- 3"? '3 ._L_._. ’ =0,oo|5 .— 7) l’YUD WV, . .T *0 EMA : E = 0.058V . Kayo ,.__.§ -: ologv F West! (@2096) C = f, [m( > (HELD F (KM; r f’mLNdi I 0 052 V,Q®'D( ZKJQ + lAL30) “7 VNa VLNWEL _ O 09$“ 0 ‘ 0.00:?) 1‘»- 0,06‘8 : _ , I‘D) ( 0.00) --r 0.002%) O 0009 v 5'3 T783 I -0.3 WV WY B13; = [N40 U3 = [K30 Ely“) [+2tfig o O 0022 M We 2 M we LIME : 124 MM. 1:! k0 Wig/rag 006) "£112 (0.069- 0 one) ":53- m 8'3 10ng / d‘L 5.! p“ K w Erlr’f Problem 2 [20 points]: A myelinated axon has an internal diameter of 10 pm, and a distance between Ranvier nodes of 1 mm. The surrounding sheath has an area capacitance of 10 uF/mz, and the electrical conductivity inside the axon is 1 (Qm)'1. at Construct a lumped electrical model of the axon where each segment is modeled as a series resistance Rs between Ranvier nodes, and a parallel capacitance Cp at the Ranvier nodes to ground (the outside of the axon). Express values for Rs and CD in terms of the above parameters. Now assume a simple model of electrical excitation at the Ranvier nodes, where the potential jumps instantaneously to 100 mV above the rest potential as soon as the potential reaches 30 mV above the rest potential. The potential then returns to rest 2 ms after excitation, and stays at rest for another 10 ms. b. Estimate the propagation speed of the action potential along the axon, in meters per second. c. Sketch the profile for the action potential at two neighboring nodes. 1. = [Mum __<,__(:1<:1 t : Career ' ; lO/AF/yf’ 3A5“ Lo'g/ = Alt [Av—BF ‘; 3mg}: [urea Cl LH : \fv- V014 + Var; v‘j’l K" it K5 K5 j Y '+ V0439. R5519 m + V l = L ' : "So/MV ’2 04k L+ 2' A v {2.56 _ {2.7 I‘UL, 32+3€F ' , Z - 7f - Z _ 2.00/A17 AC 50mY= SOmY, (L. C“T_> W 435470114X ‘2 U" ’L: ’ 1W * 5% W/4 P” bl ’ At w/w f 4f Problem 3 [15 points]: The ECoG (electrocorticogram) biopotential on the cortical surface results from volume conduction of synaptic currents over large numbers of synchronously active cortical neurons. Consider one million excitatory synapses simultaneously activated in cortex 1 cm below the skull, where at each synapse 1pA of current enters he postsynaptic cell. Another one million inhibitory synapses are simultaneously activated 1 cm deeper in cortex (2 cm below the skull), where at each synapse 1pA of current leaves the postsynaptic cell. The extracellular space is assumed uniformly conducting with average volume conductivity 0.1 (Qm)'1. a. Show that the resulting extracellular volume biopotential for this dipole is zero perpendicular to the dipole axis (the axis spanning the excitatory and inhibitory centers), and is maximum along the dipole axis. b. Find the E006 biopotential just below the skull along this dipole axis. A '> A»?! c9, \ i_, #— A.’: I 40ml 1: Loéx : I A [+£r'fsj. 0+3— Problem 4 [15 points]: A non-polarizing surface electrode with conducting electrolytic gel is placed on the left chest just right of the sternum. A reference elestrode, of the same alloy and electrolytic composition, is placed on the upper right arm, and ano‘her electrode on the right leg is driven to system ground. The signal and reference electrodes are connected to the differential input of a bandpass amplifier with midband gain of 1,000, lower corner cutoff of 0.1 Hz, and higher corner cutoff of 40 Hz. The subject is at rest with a healthy heart at 60 beats per minute. a. Sketch the time waveform that you expect to see at the output of the amplifier. Indicate the scale on the horizontal and vertical axes. Show on the waveform the onset of atrial depolarization, ventricular depolarization, and ventricular reaolarization. b. How would you detect possible coronary occlusion ischemia from the shape of the waveform? Illustrate the difference compared with the normal waveform. Problem 5 [10 points]: Two identical non-polarizing electrodes, of unknown alloy, are placed in a saline solution with uniform concentration. The impedance between the electrodes through the solution is measured, and determined to be 500 kQ at DC and 200 Q at 1MHz, with a low- pass response at 25Hz cut-off frequency. a. What would you expect for the voltage reading for a high-impedance voltmeter between the two electrodes at equilibrium, and why? b. Are you able to identify all the circuit parameters for the equivalent circuit model of the electrode in Figure 5.4? Find as many parameters as you can. If you are missing any, how would you extend the experimental setup to measure these? V. V2 / Z (oi) : itils»+£ci,):sooluz ;) We jab : 2W. 25Hz :) [+ Lips: £0 Cam #007) Mm Problem 6 [25 points]: Design an instrument that detects the onset of epilepsy from an ECoG signal recorded from an intracranial electrode, and that drives an integrated microfluidic pump for automated delivery of a therapeutical neurochemical. The intracranial electrode is near the epigenetic center, and picks up an ECoG signal in excess of 1 mV amplitude swing during epileptic seizure, and below 100 “V swing in normal brain state. You should block DC drifts in the electrodes and E006 signal using a highpass filter input to yo Jr amplifier with a 1Hz cut-off frequency. Minimize the power consumption of your design for implantable use. Your design should run off a single 3.3V battery. A google search for “micropower” will return plenty of available amplifiers, comparators, and logic components, eg. LMP2231 micropower opamp, LM06762 micropower dual comparator, and 74HCT series logic, all operating with rai—to-rail inputs over the 3.3V supply range. You may assume that the microfluidic pump activates upon a 3.3 V digital input signal, and takes negligible input current. EXW][%: WVWD mam m +3.3V -+3,3V +3.91 9W; _/— R4,, +39V (mww Rab :ZISV (n) oifia Y / .: ) 63v v, .. y ’ gm “5‘ E f V MU (de V%-/ I )DY ». . M Ra: 1&7 WV)‘ w. Aim 0 Add 12% \22 Wm ecu/«MT VWW‘WW Y a 1 ('1’ ' : CM» x” OvnoL Kurt/thy a i, W {M \ £5 K , To IET‘ECTOK C )N “"1 LMF2231 (My “f 3% {Fa/1V \ ’mpc “Wrgwwfij; (o W? A ...
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HW5_Soln - BENG 1863 Winter 2010 Homework 5 Solutions...

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