2. 2
first type is equivalent to an electrical circuit built in electronics. A neuron is modeled to
describe its electrical properties in a measurable way. This model relates parts of the
neuron to parts of a traditional electrical circuit, such as batteries or capacitors. This model
is critical to learning how neurons use electrical signals to communicate with one another.
This is the type of model I am referring to when I bring up neuromorphic circuits.
The second type of circuit can be referred to as an anatomical circuit. It is widely
used in the neuroscience community to study systems, such as the motor system or the
visual system. This specific type of circuit has no relation to a circuit in electronics. It is
simply a convenient way to talk about groups of interconnected neurons. For instance, a
basal Ganglio-Thalamocortical circuit, is describing a group of neurons which send
information from the basal ganglia to the thalamus, and then from the thalamus to the
cerebral cortex and not relating specific parts to circuit components.
Lastly, the final type of circuit I came across is known as a functional circuit.
Functional circuits may or may not be referring to the electrical properties of a cell. For
example, a working memory circuit is a type of functional circuit. One way of talking about
working memory is by studying the electrical properties (threshold value, discharge rate,
spontaneous activity, etc.) of the neurons that are related to this part of the brain to try to
understand how information in the memory is coded and transmitted. This circuit may
then also be studied by describing the anatomical connections between groups of neurons
that are thought to process information about working memory.
3. 3
Results and Discussions
The starting point for this research was to read through several papers to look for
potential experiments I could use to model my own research off. After reading a large
amount of papers, I came across three potential candidates. The first was a paper that tried
to link information theoretic (variational) and thermodynamic (Helmholtz) free-energy
formulations of neuronal processing and show how they are fundamentally related [9].
This paper also explained that biological systems will behave in a way that minimizes
changes in Helmholtz free energy and will prefer to move towards a non-equilibrium
steady state that has developed as an evolutionary result. I was interested in this paper
because I felt that the emergent behavior found in coupled ring oscillators would agree
with the fundamental behavior found in nature that this paper emphasizes on.
The second potential candidate was a paper that studied the behavior of a mapped
clock oscillator (MCO) as a ring device and considered the potential of it serving as neural
prostheses for treating dynamic diseases such as epilepsy [15]. I felt this paper’s approach
to modeling neuronal populations would correlate nicely with my own research.
The third potential candidate was a paper that was recently published in April of
2016. This paper studied the behavior of migrating monarch butterflies and was successful
in developing a model of a time-compensated sun compass used by these butterflies [10].
Through special integration of neuronal oscillations they successfully enabled corrections
to southwest and northeast flight. I considered this paper to be a potential candidate
because the neural circuit they developed in a way studied the emergent behavior found in
monarch butterflies that allows them to synchronize in a seemingly chaotic system and
4. 4
reach their destination, 4,000 km away.
After much consideration, I decided to develop three individual circuits, which I call
“Cells”, that would have unrelated frequencies and then attempt to have them communicate
to produce a new emergent pattern that behaves differently from when they are not linked
together.
The program that was chosen to develop these circuits was LT SPICE. This specific
program was chosen because a colleague of mine, Linda Gong, had experience working
with LT SPICE from a previous semester and would be able to provide me with important
resources to help me get started with my simulations. Linda provided me with the
following symbols: Single inverter, 3-gate ring oscillator, 5-gate ring oscillator, 15-gate ring
oscillator, 31-gate ring oscillator and XOR gate. After familiarizing myself with the
parameters in the symbols and verifying the symbols were working properly, I delved into
building my first cell.
My first cell was built with a 3-gate ring oscillator coupled with a 17-gate ring
oscillator as can be seen in Figure 1. In order to couple these two oscillators successfully, I
needed to use an XOR gate to prevent confusion between the different outputs of the
oscillators. The XOR gate was fed by the output of the 3-gate ring oscillator, the output from
the 15th inverter of the 17-gate ring oscillator and then the XOR’s output was fed into the
16th inverter. A Pulse function was used as a voltage source in order to start the
oscillations. The Pulse function was set to start with an initial value of 0 volts, and with an
ON state set to 1.5 volts. The function was set to have no delay, rise or fall time. The ON
time of the Pulse was set to 0.5 microseconds. I specified a period of 10 microseconds and
5. 5
500 cycles to keep the simulation from running forever.
I also included a, cmosedu, text file that was provided by Dr. Jacob Baker, a
professor at the University of Nevada. This text file contains parameters for various
components of the Monolithic P-MOS and N-MOS models (internal capacitance, resistance,
etc). For my simulation I performed a transient analysis to determine how the circuit will
behave under non-well-behaved signals. By checking if my circuit becomes unstable under
certain conditions, I can determine it is not a robust circuit. In addition, I bypassed the DC
operating point analysis in my transient analysis by including the UIC (to calculate the
initial transient conditions rather than solving for the quiescent operating point)
parameter.
Figure 1: “Cell 1” 3-gate RO coupled with 17-gate RO
6. 6
The simulation begins with a transient state and then stabilizes at around 0.6
microseconds. The first interference pattern starts at 1.2 microseconds and stops at about
2.0 microseconds. Interference patterns will continue to appear about every 1.2
microseconds and last for about 0.7 microseconds. Figure 3 shows the different waveforms
found in Cell 1 in more detail. The center waveform seems to resemble the I-V
characteristics found in a capacitor.
Figure 2: Simulation of Cell 1 for 5.5 microseconds
9. 9
Finally, I built the largest Cell of all three using the same procedures as Cell 1 and
Cell 2. Cell 3 was designed with a 17-gate ring oscillator coupled with a 49-gate ring
oscillator as can be seen in Figure 7. The behavior of Cell 3 is shown in Figure 8. As
expected, Cell 3 behaves differently compared to Cell 1 and Cell 2. A transient state is still
observed in Cell 3, however it is far more difficult to observe the interference pattern. To
show the interference pattern I provided a zoomed in cut out of the first 2.2 microseconds,
shown in Figure 9. The interference lasts for about 0.8 microseconds. Cell 3 has a much
larger frequency compared to Cell 1 and Cell 2.
Figure 6: Closer look at the waveforms found in Cell 2
11. 11
The next objective was to have all three Cell’s communicate with each other. To
accomplish this, I arranged a series configuration of all three Cell’s. Cell 1 was fed into Cell 2
and Cell 2 was fed into Cell 3 as can be seen in Figure 10. I had some difficulty finding the
right parameters to use for the Pulse function. After testing a few different parameters I
found the simulations worked best if the function was set to have a longer ON time than the
larger ring-oscillators. The initial and ON state were still set to 0 volts and 1.5 volts
respectively and the function was still set to have no delay, rise or fall time. However, the
ON time of the Pulse was set to 1 microsecond instead of 0.5 microseconds. I also specified
a period of 2 microseconds and 1000 cycles. The transient analysis was set to 1000
microseconds.
Figure 9: Simulation of Cell 3 zoomed in to 2.2 microseconds
12. 12
The final circuit produced an interesting behavior. As can be seen in Figure 11, the
circuit experienced a shorter transient state and began to stabilize below 0 volts. I believe
this occurred as a result of inserting signals with much smaller frequencies into signals
with much larger frequencies. The circuit appears to rise above 0 volts at about 1.13
microseconds. A clearer observation of this is shown in Figure 12. In addition, one can
clearly observe to different waveforms throughout the simulation. Figure 12 seems to
resemble modulation found in telecommunications. The interference pattern of this circuit
appears to begin at approximately 1.2 microseconds and last for about 0.8 microseconds.
Figure 10: Series configuration of all three Cell’s
15. 15
Future Work
Due to time I was not able to test more circuits that would produce interesting
emergent behavior. Figure 14, shows a circuit I was working on where I was including a
pass-gate that would be switched on and off using another ring oscillator. I would like to
get this circuit working and then build similar circuits and test different configurations to
see what sorts of behaviors I may find. My goal for future research would be to find a
potential model that can be imbedded onto an electronic circuit and can then be used as a
potential neural prosthesis to help treat patients with epilepsy or to build better computer
architectures that resemble more closely to how our own brains compute and transfer
information.
Figure 14: 17-gate RO coupled with 49-gate RO with pass-gate
17. 17
References
1. Canavier, C.c., R.j. Butera, R.o. Dror, D.a. Baxter, J.w. Clark, and J.h. Byrne. "Phase
Response Characteristics of Model Neurons Determine Which Patterns Are
Expressed in a Ring Circuit Model of Gait Generation." Biological Cybernetics 77.6
(1997): 367-80.
2. Curtis, Clayton E., and Mark D'esposito. "Persistent Activity in the Prefrontal Cortex
during Working Memory." Trends in Cognitive Sciences 7.9 (2003): 415-23.
3. Dror, R. O., C. C. Canavier, R. J. Butera, J. W. Clark, and J. H. Byrne. "A Mathematical
Criterion Based on Phase Response Curves for Stability in a Ring of Coupled
Oscillators." Biological Cybernetics 80.1 (1999): 11-23.
4. Fuster, Joaquín M. "Memory." Cortex and Mind (2005): 111-42.
5. Graf, H. P., L. D. Jackel, R. E. Howard, B. Straughn, J. S. Denker, W. Hubbard, D. M.
Tennant, and D. Schwartz. "VLSI Implementation of a Neural Network Memory with
Several Hundreds of Neurons." AIP Conference Proceedings (1986): n. pag.
6. Guertin, Pierre A. "Central Pattern Generator for Locomotion: Anatomical,
Physiological, and Pathophysiological Considerations." Frontiers in Neurology Front.
Neur. 3 (2013): n. pag.
7. Hagler, Donald J., and Martin I. Sereno. "Spatial Maps in Frontal and Prefrontal
Cortex."NeuroImage 29.2 (2006): 567-77.
8. Jungblut, Melanie, Wolfgang Knoll, Christiane Thielemann, and Mark Pottek.
"Triangular Neuronal Networks on Microelectrode Arrays: An Approach to Improve
the Properties of Low-density Networks for Extracellular Recording." Biomedical
Microdevices Biomed Microdevices 11.6 (2009): 1269-278.
9. Sengupta, Biswa, Martin B. Stemmler, and Karl J. Friston. "Information and Efficiency
in the Nervous System—A Synthesis." PLoS Comput Biol PLoS Computational
Biology9.7 (2013): n. pag.
10. Shlizerman, Eli, James Phillips-Portillo, Daniel B. Forger, and Steven M. Reppert.
"Neural Integration Underlying a Time-Compensated Sun Compass in the Migratory
Monarch Butterfly." Cell Reports 15.4 (2016): 683-91.
11. Srimal, Riju, and Clayton E. Curtis. "Persistent Neural Activity during the
Maintenance of Spatial Position in Working Memory." NeuroImage 39.1 (2008):
455-68.
12. Sussillo, David, Paul Nuyujukian, Joline M. Fan, Jonathan C. Kao, Sergey D. Stavisky,
Stephen Ryu, and Krishna Shenoy. "A Recurrent Neural Network for Closed-loop
Intracortical Brain–machine Interface Decoders." J. Neural Eng. Journal of Neural
Engineering 9.2 (2012): 026027.
13. Vishwanathan, Ashwin, Guo-Qiang Bi, and Henry C. Zeringue. "Ring-shaped
Neuronal Networks: A Platform to Study Persistent Activity." Lab on a Chip Lab
Chip 11.6 (2011): 1081.
14. Volman, Vladislav, Richard C. Gerkin, Pak-Ming Lau, Eshel Ben-Jacob, and Guo-Qiang
Bi. "Calcium and Synaptic Dynamics Underlying Reverberatory Activity in Neuronal
Networks." Phys. Biol. Physical Biology 4.2 (2007): 91-103.
15. Zalay, O.c., and B.l. Bardakjian. "Mapped Clock Oscillators as Ring Devices and Their
Application to Neuronal Electrical Rhythms." IEEE Trans. Neural Syst. Rehabil. Eng.
