Lecture 10 Chapter 5 February 6, 2019

Biological Amplifiers 1 A few molecules can trigger the release of 10,000 Ca++ ions. 2. A small voltage can open channels at a gap junction so that voltage gain can occur for current flowing from a large cell to a small one with a larger resistance. 3. Most biological systems have negative feedback to help stabilize the system. 4. For temperature control G≈-33 For blood pressure -2

Biological Amplifiers http://www.unmc.edu/physiology/Mann/mann13.html Neural Transmitter Releases up to 104 calcium ions Need to overcome the electrical threshold for firing

Electronic Amplifiers 1 Basic Amplifier use energy from one source and use it to increase the strength of the desired signal. 2. Electronics we take energy of a DC power supply and use it to increase the amplitude of the desired signal. 3. This is not essential we can get our energy from an AC signal in a parametric amplifier or from noise in stochastic resonate amplifier

Oscillators Two approach to analysis 1 Negative resistance oscillator R R C L 2 Feedback oscillators

Basic Feedback Oscillator Vi= Vs+βVo Vo=AVi Vo[1/A-β]=Vs Vs + Vi A Vo β

Operational Amplifier with Time Delay in the Feedback

Steady State Solution for Vs= Vin cos(ωt) and Vo cos (ωt –θ) where θ=ωτ, = Note change in sign with θ=ωτ so we can get either amplification or attenuation by changing frequency or the time delay τ

Oscillation The system breaks into oscillation when the gain As the gain Af oscillates from zero to

NADPH, ROS, NOS Oscillations NAD(P)H concentration in motile neutrophils is oscillatory, and the amplitude of the oscillation can resonate in the sense that the amplitude increases with externally applied pulsed magnetic fields. NAD(P)H autofluorescence was monitored with a photomultiplier, and photomultiplier counts plotted . Note the amplitude of the signal returns to its normal value when the stimulus is removed. Rosenpire et.al (2005) Figure 2.

Redox Oscillation Reduction Figure 3. Flavoprotein redox oscillations are inhibited by pulsed magnetic fields timed to coincide with minimal flavoprotein autofluorescence and they amplify the oscillations when timed at the minimums. Rosenpire et.al (2005) Figure 11.

Self Limiting Protein Synthesis Suppose that the rate of protein synthesis at present (at time t) depends on the concentration of protein at some time in the past (at time t-τ), where τ is the time delay required for transcription and translation. Then, the governing kinetic equation becomes (2) Béla Novak* and John J. Tyson ” Design Principles of Biochemical Oscillators Nat Rev Mol Cell Biol. 2008 December ; 9(12): 981–991. doi:10.1038/nrm2530.

Effects of Time Delay Between E and J This can give Z in all four quadrants.

Nonlinear Elements 1. A nonlinear resistance 2. A nonlinear reactance 3. A time varying element in you circuit or system. 4. These elements show up in many form and the biological ones are more complicated than the electronic ones.

Basic Characteristic of Nonlinear Devices. 1. Nonlinear resistance,

Semiconductor Diode The simple one is a diode. I= Vo+αV1+βV2+---- I I

An Ideal Harmonic Generator 1 Two tunnel diodes in series

Test Circuit

Results

Nonlinear Reactance 1. Use to convert power from one frequency to another. 2 Typical diode C~(V)-1/2 for step diode 3 How do you design a diode with a larger nonlinear capacitance? P-_ N_ P+ N+ Ni N+

Capacitors C = Q/V C= εA/d V= QC=Qd/εA

Parametric Amplifiers 1. Conservation of Energy on a photon basis 2. Conservation of momentum where k is the propagation constants

Parametric Amplifiers

Lecture 10 Reference on Stochastic Resonance “Tuning in to Noise” Adi Bulsara and Luca Gammaitoni Physics Today March 1996 Stochastic Resonance addition to BEMS paper Kendra Krueger October 2011

Stochastic Resonance

Stochastic Resonance

S/N Ratio vs N 1 SNR External Noise Intensity Optimal noise level

Current Flows.

Concentration of Electric Fields in Space The voltage across the membrane becomes approximately equal to V/2 and the Em ≈ L/2t

Nonlinear Effects at Cell Membranes Current flow for Rm is the membrane resistance. The result is that the membrane is a poor rectifier. However AC voltages make the interior more negative.

AC Induced Current Flows At Low Frequencies Induced DC Currents for VAC from -60 to + 40mV For a spherical cell.

Shift in Membrane Firing Time Shift in firing time for Where u(t) is unit step function

Mode Locking of Oscillators Theory for injection locking of electronic oscillators is give by The theory is good for case where This worked for Aplysia pacemaker cells.

Signal Noise Requirements for Phase Locking The phase of the inject signal must be stable enough so that the phase φ Where K is the linear control characteristic in units (2π Hz/V) and is closely related to the loop gain.

Threshold Injection Locking for an Aplysia Pacemaker Cell Frequency range from 2 to 10 Hz

Locking of a Pacemaker Cell Response to various frequencies of injected currents.

Coherence 1.Coherence in Space and Time. 2. Neural Networks and Learning 3. Source of Noise and Minimum Detectable Signals ?

Signal Coherence Litovitz showed that for 10µT coherence for 10 seconds or longer was required for signals at 55 or 65 Hz was required to change the activity of τcell= 8 sec

Litovitz shows both space and time coherence help separate signals from Noise

Results Show 1. Both Space and time Coherence are important. 2. Small electric fields can lead to biological changes. 3. Magnetic fields can affect biological changes by a separate mechanism.

Membrane Capacity as a Function of Frequency Membrane Capacity is only a slowly varying function of frequency.

A Neural Network Model for Adaptive Responses 1

Training to Recognize 60Hz as a Function of S/N with 97% Accuracy

Lecture 11

Natural and Man-Made Fields 1. The atmosphere charged about 100/sec world wide with about an 18 sec time constant to about 130V/m 2. Peak values at about 3000V/m 3. Rapid decrease with frequency to typical value > 1 Hz of 10-4 V/m 4. These numbers are all variable

Internal Fields 1. Across a membrane of 2 x 107V/m 2. Nerve pulses about 0.4ms , rise time 0.1ms fall time 0.5ms. Dead space 1 to 3ms 3. Fields along the outside of a nerve cell 5x10-2V/m 4. These numbers are variable with position, type of cell etc.

Types of Noise 1. Thermal 2. Shot Noise 3. C/fn Noise 4. Noise generated by other electrical activity in the Body.

Thermal Noise. 1. Black body radiation For h f <<kT Pn= kTB = kTΔf 2. Other forms for matched loads 3 For thermal equilibrium. Non-equilibrium get negative temperatures.

Spontaneous Emission and Shot Noise P= hfΔf 2. Shot Noise 3. 1/f Noise or Where S(f) is the power spectral density

Example 1/f noise for a hole in a mylar film 1. For mylar film b is a geometrical factor a is a constant r is the radius of the hole Φ is the applied voltage

Membrane Example. 1

Other Electrical Activity 1. EEG 2. ECG or EKG 3 Muscle movement. 4. Nerve Cells Firing

Minimum Detectable Electric Field Is a Function of Frequency Bovine Fibroblast Cells I= 10-3—10A/m2