2. Foundations
Neural nets are a mainstay of artificial intelligence research, the models that will be used here are derived from
the earliest work in the field, the McCulloch-Pitts Cells of 1943.1 Neural nets in general excel at certain types of
computation, such as pattern recognition, which are very difficult to perform using other methods. They also possess
the unique ability to learn through evolution, so that they become extremely efficient in handling a given task.
At their most basic, neural nets are formed by neurons, and the connections between them. Neurons receive input
signals from other neurons, and output a signal to other neurons. Each neuron possesses a quality known as its
threshold of activation, which is the level of input required to make the neuron output a signal in response. The many
possible combinations of inputs, thresholds, and outputs amongst neurons in a network result in a system that is
capable of great complexity.
In recent years, electronic musical equipment has begun to appear that
operate on metaphors of neural networks, and this emergence into my field
of expertise has finally produced an environment from which I can make
meaningful inquiries. The systems relevant to this proposal are not capable
of learning, but they do base their operation on the unique neural net
properties of threshold activation, interconnectedness, and signal feedback.
The strategies of implementation and degrees of artistic licence vary
between designers, and the differences are quite radical between the two
examples that are currently implemented in analog hardware: the Resonator
Fig. 1 Michaelis Resonator Neuronium, by Jürgen Michaelis,2 and the Wiard Neural Pulsers, by Grant
Neuronium, 2005 Richter.3
The Resonator Neuronium (Figure 1) is a complete analog synthesizer
that uses neural circuitry to generate all of its sound-shaping functions.
Its complex architecture weaves together the traditional disparate building
blocks of audio synthesis into a larger symbiotic system, where every element
has a degree of influence over every other.
In contrast, the Wiard Neural Pulsers (Figure 2) is intended as a single,
specialized module in a far larger modular synthesis system. Instead of
dealing with smoothly-varying signals, the Neural Pulsers operate with pulses,
like in Figure 3 – abrupt transitions in voltage lasting about 10 milliseconds.
Fig. 2 Wiard Neural This type of signal is often called a “trigger,” because of its common use in
Pulsers, 2003 analog synthesizers as a timing function, triggering the start of events within
the system. Therefore, the primary use of the Neural Pulsers is as a multi-
part rhythm generator. It also interfaces easily with the many sensor systems
that can interpret triggers, such as those commonly used for interactive
10V performance and installation.
While the designers of both systems have documented their efforts online,
The Neural Pulsers is the much simpler, and hence more approachable
circuit. It’s also directly signal-compatible with the modular equipment that
I have. Therefore, I chose to prototype and analyze the Neural Pulsers as
a beginning for my explorations. What I will propose then is a rethinking of
0V
10mS Richter’s research into the development of experimental hardware neurons
for music synthesizers, yeilding a device whose functionality is an order of
Fig. 3 A pulse waveform magnitude greater than the previous implementation.
2
3. The Wiard Neural Pulsers
Richter’s Neural Pulsers circuit consists of two distinct EIGHTH QUARTER HALF WHOLE
parts: a master timing clock, and the neurons themselves.
The clock defines discrete time – the synchronous
moments at which the neurons’ states are updated – and
simultaneously provides the logic to drive the neurons. Fig. 4 The divided clock outputs
Figure 4 shows how the clock is divided, with 4 outputs of the Neural Pulsers.
whose divisions correspond to musical whole, half, quarter,
and eighth notes. The eighth note output is actually the
clock itself, and therefore the smallest unit of time in the
circuit. In most cases, at least one of these outputs is
patched to the neurons, and usually two or more are used in
combination. The use of banana jacks for patching makes it 2 OUT
easy to distribute the signal from a single output to multiple
inputs, so that any one neuron is capable of influencing all EXC 0
of the others. The use of discrete time in the circuit imposes
a unique characteristic, and that is the ever-present logic 1
delay. A neuron that receives input at time t cannot act upon
that input until time t + 1. This fundamental property can be
exploited, and must always be kept in mind when analyzing INH
a patch’s state.
Figure 5 details the core of the circuit, which is the 4 identical Fig. 5 The panel layout of a single Wiard
McCulloch-Pitts neurons. These exist in the simplest form neuron, shown at actual size.
possible, with two equally-weighted excitatory inputs, and
a single overriding inhibitory input. Inputs are excitatory if
the signals coming into them contribute positively toward
reaching the threshold – the level at which the neuron will
“fire,” and send a signal from its output. Inhibiting inputs
have the opposite effect, reducing the overall input level. To summarize the threshold logic:
The value of all inputs are considered simultaneously, and
weighting describes the relative influence of one input in • A neuron with threshold 2 (AND) needs two
comparison to another. Equally-weighted inputs of the same excitatory signals and no inhibitory signal
type are interchangable. The inhibitory input in this design present at time t in order to fire at t + 1.
is not equally-weighted to the excitatory inputs, but is said
to be overriding because irregardless of what other signals • A neuron with threshold 1 (OR) needs only
are present, a signal at this inhibitory input will disable the a single signal, at either of the excitatory
neuron’s output at the next clock cycle. inputs, with no inhibitory signal present at
time t in order to fire at t + 1.
The threshold of a neuron in the Neural Pulsers is set by a
3-position switch to the level of 0, 1, or 2. This arrangement • A neuron with threshold 0 (NOT) will always
defines the Boolean logic gates NOT, OR, and AND, fire at time t + 1, unless there was an
respectively. A logical TRUE state results if the combination inhibitory signal present at time t.
of pulses at the inputs satisfies the selected function,
causing the neuron to fire, and output a pulse at the next • This also determines that a threshold 0
clock. This pulse appears at a status indicator LED, and neuron wlll not fire at the first clock cycle
at two identical outputs – one banana jack, for patching of a patch’s operation, because the neuron
back into the system, and one 3.5mm jack, for driving any always takes a full clock cycle to check its
external device that can respond to +10V triggers. inputs before it can process them.
3
4. The neuron itself is situated between the clock outputs
and its I/O graph, with its three input jacks on the left,
and the sole output jack on the right. The threshold of
the neuron is printed directly upon it, and this is the
first thing to note, since it will factor in all remaining
calculations. For Figure 7’s neuron to fire at the next
clock cycle (neurons always require 1 clock cycle to
Understanding the Diagrams to Follow process their inputs), there must be no inhibitory pulse
present, and the sum of its excitatory inputs must be
A patch is formed by interconnecting any of the available greater than or equal to 1.
clocks, inputs, and outputs. The I/O diagrams chart all
relevant details of a patch over the course of time. The At step 1, the neuron has 1 inhibitory pulse, and 1
timelines span 17 clock cycles (eighth notes) in order excitatory pulse at its inputs. Because of the inhibitory
to show a patch’s initial state, as well as to indicate pulse, the neuron will not fire at the next clock cycle,
what happens when the logic settles into a loop. I/O and indeed, there is no output pulse indicated at step
points are color-coded, unfilled circles, corresponding 2 of the graph. Step 2 has no inputs at all, and since
to the real-world banana jacks that they represent. this neuron needs at least 1 to fire, there will also be
Clock and neuron outputs are red, excitatory inputs are no output from the neuron at step 3. At step 3, a single
blue, and inhibitory inputs are grey. Connections made excitatory pulse arrives, and unlike step 1, there is no
by patch cables are indicated in black. inhibitory pulse present. Therefore, at step 4 the neuron
finally fires an output pulse in response! This simple
A path extends from every jack onto the I/O graph, with pattern then repeats indefinitely.
one exception: the clock ouputs. The clock’s output
patterns never change, and so they are assumed In this example, no other neurons were used, but the logic
present to reduce clutter. If they were mapped onto the is the same no matter how many are interconnected.
grid, they would appear as in Figure 6. A solid circle The essential rule to remember is that, while pulses are
indicates each time a signal appears at a jack. The transferred instantaneously from output to input, there
clock output jacks will normally be shown unmapped, is always a single clock cycle delay required to process
positioned to the left of the neurons, as in Figure 7. inputs into an output pulse.
1 1 1 5 9 13 17
1 5 9 13 17
½ ½
1
¼ ¼
⅛ ⅛
Fig. 6 Visualization of the clock outputs Fig. 7 A single Wiard neuron driven by two of the clock
outputs
4
5. Working with the Wiard Neural Pulsers
1 5 9 13 17
Despite multiple neurons and patching possibilities,
the Wiard module turns out to be a very simplistic A 0
device. As configured, it’s capable of little more than
1
the most basic output sequences. The central role
of the clock in rhythm generation makes it nearly ½
B 1
impossible to create irregular patterns. The most
¼
complex output is realized through the techniques
of signal duplication, delay, and feedback. C 1
⅛
Signal duplication has many uses, such as routing D 1
the output of neuron A into both the excitatory
input of neuron B and the inhibitory input of neuron
C. The firing of A will then make B and C operate
very differently at precicely the same moment. Fig. 8 Canonic pulse train of length n, followed by n stages
of inactivity, where n is the number of neurons used.
Delay is induced by routing a signal into an
excitatory input of a neuron with threshold 1, and
1 5 9 13 17
then taking the identical, but one-cycle-behind
output. It can be used to unbalance the otherwise
static outputs from the clock, when mixed together A 1
through another neuron. 1
½
B 1
Feedback occurs when a neuron’s output is routed
into one of its inputs. As pointed out by Minsky ¼
in his analysis of McCulloch-Pitts, this creates a C 1
1-bit memory, where a neuron’s firing state at time ⅛
t depends upon its firing state at time t - 1.4
D 1
Another of the Neural Pulsers’ strengths is the
external output available from each neuron, which
makes them capable of driving 4 discrete voices, Fig. 9 Creating an irregular pattern at D by using B as a
for example. If delay is implemented serially in delay.
each of the 4 neurons, the logical equivalent of a
musical canon is produced, as in Figure 8.
5
6. Advancing the Design of the Neural Pulsers
I initially thought that the most significant limitation of the Wiard Neural
Pulsers was the small number of neurons used. A simple delay function
requires one neuron per stage, and functions of any complexity use up
the 4 neural elements quickly. Conversely, it is generous to describe
any function available from 4 neurons in this implementation as
complex. It seemed that the logical answer was simply more neurons,
and I estimate that having 10 or so would yield greater possibilities.
Still, I don’t imagine that the brute addition of 6 or more neurons would
make this system into more than the sum of its parts – what I imagine
instead is more of the same. With 36 faceplate elements already, the
Fig. 10 Reconfigured Input physical and financial burdens of this method of enhancement rule it
stage, featuring a pair out.
of subtracting inhibiting
inputs. What I’ve done instead is to reevaluate the architecture of the system.
Having read Minsky’s excerpt on McCulloch-Pitts cells, I already knew
that other implementations were possible. I thought through several
uninspiring alternatives before arriving at my solution, which completely
reconfigures the input, threshold, and output stages, leaving behind
+2 most of the unneccesary restrictions of Richter’s design.
+1
0
-1 The first enhancement is to reconfigure the neuron input stage. The
Wiard neurons suffer from having their only inhibitory input be one with
total precedence. From this point on, I’ll refer to such overriding behavior
as a disabling input, because I see enormous potential in introducing
a pair of inhibitory inputs which merely subtract from the input value,
Fig. 11 Reconfigured Threshold rather than incapacitate a neuron entirely. Sharing equal weighting with
and Output stages, with the existing adding excitatory inputs, the new subtracting inhibitory
synchronous discrete inputs would expand the range of incremental change available at the
outputs for each input summer. Five inputs configured as in Figure 10 can present the
possible threshold. levels {+2, +1, 0, -1, -2} and the logical NOT to the threshold comparator.
6
7. This is two more levels than the threshold switch is capable of
distinguishing, so the switch now becomes a bottleneck, limiting the
expanded sensitivity of the new input stage. A switch with more poles
could take its place, but the dilemna importantly reveals that there is
no benefit in maintaining the existing threshold/output architecture,
where a single comparison produces a single output. Figure 11 shows
that if instead, each of the 4 possible thresholds {+2, +1, 0, -1} are
+2
given synchronous, independent logic outputs, the traditional threshold
mechanism can be discarded, leaving an open and far more flexible +1
logic device. The neural structure that results from the above changes 0
is so much more powerful than the original Wiard design, that a single -1
neuron of the proposed format can rival the functionality of the 4 that
were previously used.
The great amount of I/O points in the proposed neuron require a
subsequent expansion of the visual feedback system, to clarify the
complex interconnections available. The output LEDs in the Neural
Pulsers are perfect at this task, and my implementation will assign one Fig. 12 Proposed neuron core,
to every I/O point, in addition to a central clock LED. The status of every with LED status indicators.
part of the module can then always be known, making the task of logic Shown at actual size.
routing an intiutive one.
The final enhancements upgrade the clock, adding 3.5mm jacks for each
clock division output, as well as one for an external clock input. These
simple additions complete the integration of the neuron with external
equipment, allowing the neuron to be synchronized to the timing of
another device as either master or slave. I can also be then driven by
non-regular pulse patterns generated from elsewhere. The clock rate
should also be expanded into the audio range, so that neurons’ pulse
outputs can be used as harmonic generators.
1 5 9 13 17
1
Fig. 13 Irregularity.
½ 2
1
0
¼ -1
⅛
1 5 9 13 17
1
Fig. 14 Generating an incremental
½ 2 increase in threshold from
1
0 minimum to maximum,
¼ -1
then resetting and
⅛ repeating.
7
