Chapter 5 Lecture 10
Spring 2015
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,
An Ideal Harmonic Generator
• 1 The simple one is a diode. I= Vo+αV1+βV2+--• V
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+
Parametric Amplifiers
• 1. Conservation of Energy on a photon basis
• 2. Conservation of momentum where k is the
propagation constants
Parametric Amplifiers
Biological Amplifiers
• http://www.unmc.edu/physiology/Mann/man
n13.html
• Neural Transmitter Releases up to 104 calcium ions
• Need to overcome the electrical threshold for firing
Stochastic Resonance
Stochastic Resonance
12
Nonlinear Effects at Cell Membranes
1. Current flow for
2. 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.
Threshold Injection Locking
for an Aplysia Pacemaker Cell
• Frequency range from 2 to 10 Hz
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.
Locking of a Pacemaker Cell
• Response to various frequencies of injected
currents.
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
 cell  8 sec
• 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.
Effects of Time Delay Between
E and J
• This can give Z in all four quadrants.
Membrane Capacity as a Function of
Frequency
• Membrane Capacity is only a small function of
voltage.
Repetitive Stimulation
• 1. Repetitive microwave pulse resulted in
decreasing the amount of slowing for a
pacemaker cell in Aplysia.
• 2. Repetitive electrical stimulation lead to
decreases in the resistance of gap junctions
and to a 62% increase in coupling between
cells.
• 3. These are likely to be the result of feedback
leading to adaptive responses.
A Neural Network Model
for Adaptive Responses
• 1
Training to Recognize 60Hz as a
Function of S/N with 97% Accuracy
Thermal Calculations
• Power in and rate of change of temperature
Maximum Temperature change for a small
sphere with total energy in H
Thermal Chemistry
•
•
•
•
S = fraction that under gone chemical change
K’ is the chemical reaction rate.
R’ is the gas constant
H’ is free energy , S’ is the entropy.
• This leads to an exponential of an exponential
Thermal Chemistry
• 1. Rule of thumb we are likely to see biological
changes when
• 2. The body typically holds your temperature
to +/- 0.5oC
• 3. Very rapid changes in chemical reaction
rates above a threshold. !!
The Rate of Change of Temperature is
also Important.
• 1 We have shown the changes of 1/10 oC can
change the firing rate of a pacemaker cell at
1oC/sec.
• From the Nernst Equation
• Slow increases in T increased firing rates of a
pacemaker cell rapid one decreased it. Changes
seen with as little as 0.1 o C at rates of 1o C/sec
Effects of Rapid Heating
• Picture from
• Aplysia
Discussion
• 1. It takes high powers and short pulses to get
significant temperature differences on small
objects.
• 2. Thin films have larger surface to volume ratios
and cool faster than spheres .
• 3. Blood flow cools hot spots.
• 4. The thermal time constant is an important
parameter and the sensitive to temperature
change is one of the first measurements to make
on any experiments involving RF or Microwaves.
Discussion
• 1. Temperature pulses lead to thermal
expansion and can cause acoustic waves that
can be sensed at a distance.
• 2. Example radar hearing.
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. Pn= kTB = kTΔf
• 2. Other forms for matched loads
• 3 For thermal equilibrium. Non-equilibrium
get negative temperatures.
Spontaneous Emission and Shot Noise
• 1 Spontaneous Emission
•
P= hfΔf
• 2. Shot Noise
• 3. 1/f Noise or
• Where S(f) is the power spectral density
Example
• 1. For mylar film
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