Figure 1.
A) A first replenished pressurized container is allowed to diffuse into two non-pressurized empty containers
and
though a region of matter M. B) The gradient
reduces faster than the gradient
due to the conductance differential. C) This causes
to grow more than
, reducing the conductance differential and leading to anti-Hebbian learning. D) The first detectable signal (work) is available at
owing to the differential that favors it. As a response to this signal, events may transpire in the environment that open up new pathways to particle dissipation. The initial conductance differential is reinforced leading to Hebbian learning.
Figure 2.
Attractor states of a two-input AHaH node.
The AHaH rule naturally forms decision boundaries that maximize the margin between data distributions (black blobs). This is easily visualized in two dimensions, but it is equally valid for any number of inputs. Attractor states are represented by decision boundaries A, B, C (green dotted lines) and D (red dashed line). Each state has a corresponding anti-state: . State A is the null state and its occupation is inhibited by the bias. State D has not yet been reliably achieved in circuit simulations.
Figure 3.
Universal reconfigurable logic.
By connecting the output of AHaH nodes (circles) to the input of static NAND gates, one may create a universal reconfigurable logic gate by configuring the AHaH node attractor states (). The structure of the data stream on binary encoded channels
and
support AHaH attractor states
(Figure 2). Through configuration of node attractor states the logic function of the circuit can be configured and all logic functions are possible. If inputs are represented as a spike encoding over four channels then AHaH node attractor states can attain all logic functions without the use of NAND gates.
Table 1.
Spike logic patterns.
Figure 4.
A differential pair of memristors forms a synapse.
A differential pair of memristors is used to form a synaptic weight, allowing for both a sign and magnitude. The bar on the memristor is used to indicate polarity and corresponds to the lower potential end when driving the memristor into a higher conductance state. and
form a voltage divider causing the voltage at node y to be some value between
and
. When driven correctly in the absence of Hebbian feedback a synapse will evolve to a symmetric state where
V, alleviating issues arising from device inhomogeneities.
Figure 5.
AHaH 2-1 two-phase circuit diagram.
The circuit produces an analog voltage signal on the output at node y given a spike pattern on its inputs labeled ,
,
. The bias inputs
,
,
are equivalent to the spike pattern inputs except that they are always active when the spike pattern inputs are active. F is a voltage source used to implement supervised and unsupervised learning via the AHaH rule. The polarity of the memristors for the bias synapse(s) is inverted relative to the input memristors. The output voltage,
, contains both state (positive/negative) and confidence (magnitude) information.
Figure 6.
Circuit voltages across memristors during the read and write phases.
A) Voltages during read phase across spike input memristors. B) Voltages during write phase across spike input memristors. C) Voltages during read phase across bias memristors. D) Voltages during write phase across bias memristors.
Table 2.
Memristor conductance updates during the read and write cycle.
Figure 7.
Generalized Metastable Switch (MSS).
An MSS is an idealized two-state element that switches probabilistically between its two states as a function of applied voltage bias and temperature. The probability that the MSS will transition from the B state to the A state is given by , while the probability that the MSS will transition from the A state to the B state is given by
. We model a memristor as a collection of
MSSs evolving over discrete time steps.
Table 3.
General memristive device model parameters fit to various devices.
Figure 8.
Generalized memristive device model simulations.
A) Solid line represents the model simulated at 100 Hz and dots represent the measurements from a physical Ag-chalcogenide device from Boise State University. Physical and predicted device current resulted from driving a sinusoidal voltage of 0.25 V amplitude at 100 Hz across the device. B) Simulation of two series-connected arbitrary devices with differing model parameter values. C) Simulated response to pulse trains of {10 μs, 0.2 V, −0.5 V}, {10 μs, 0.8 V, −2.0 V}, and {5 μs, 0.8 V, −2.0 V} showing the incremental change in resistance in response to small voltage pulses. D) Simulated time response of model from driving a sinusoidal voltage of 0.25 V amplitude at 100 Hz, 150 Hz, and 200 Hz. E) Simulated response to a triangle wave of 0.1 V amplitude at 100 Hz showing the expected incremental behavior of the model. F) Simulated and scaled hysteresis curves for the AIST, GST, and WOx devices (not to scale).
Figure 9.
Unsupervised robotic arm challenge.
The robotic arm challenge involves a multi-jointed robotic arm that moves to capture a target. Each joint on the arm has 360 degrees of rotation, and the base joint is anchored to the floor. Using only a value signal relating the distance from the head to the target and an AHaH motor controller taking as input sensory stimuli in a closed-loop configuration, the robotic arm autonomously learns to capture stationary and moving targets. New targets are dropped within the arm’s reach radius after each capture, and the number of discrete angular joint actuations required for each catch is recorded to asses capture efficiency.
Figure 10.
The AHaH rule reconstructed from simulations.
Each data point represents the change in a synaptic weight as a function of AHaH node activation, y. Blue data points correspond to input synapses and red data points to bias inputs. There is good congruence between the A) functional and B) circuit implementations of the AHaH rule.
Figure 11.
Justification of constant weight conjugate.
Multiple AHaH nodes receive spike patterns from the set while the weight and weight conjugate is measured. Blue = weight conjugate (
), Red = weight (
). The quantity
has a much lower variance than the quantity
over multiple trials, justifying the assumption that
is a constant factor.
Figure 12.
Attractor states of a two-input AHaH node under the three-pattern input.
The AHaH rule naturally forms decision boundaries that maximize the margin between data distributions. Weight space plots show the initial weight coordinate (green circle), the final weight coordinate (red circle) and the path between (blue line). Evolution of weights from a random normal initialization to attractor basins can be clearly seen for both the functional model (A) and circuit model (B).
Figure 13.
AHaH attractor states as logic functions.
A) Logic state occupation frequency after 5000 time steps for both functional model and circuit model. All logic functions can be attained directly from attractor states except for XOR functions, which can be attained via multi-stage circuits. B) The logic functions are stable over time for both functional model and circuit model, indicating stable attractor dynamics.
Table 4.
Logic functions.
Table 5.
AHaH clusterer sweep results.
Figure 14.
Functional (A) and circuit (B) simulation results of an AHaH clusterer formed of twenty AHaH nodes. Spike patterns were encoded over 16 active input lines from a total spike space of 256. The number of noise bits was swept from 1 (6.25%) to 10 (62.5%) while the vergence was measured. The performance is a function of the total number of spike patterns. Blue = 16 (100% load), Orange = 20 (125% load), Purple = 24 (150% load), Green = 32 (200% load), Red = 64 (400% load).
Figure 15.
Two-dimensional spatial clustering demonstrations.
The AHaH clusterer performs well across a wide range of different 2D spatial cluster types, all without predefining the number of clusters or the expected cluster types. A) Gaussian B) non-Gaussian C) random Gaussian size and placement.
Table 6.
Benchmark classification results.
Figure 16.
Classification benchmarks results.
A) Reuters-21578. Using the top ten most frequent labels associated with the news articles in the Reuters-21578 data set, the AHaH classifier’s accuracy, precision, recall, and F1 score was determined as a function of its confidence threshold. As the confidence threshold increases, the precision increases while recall drops. An optimal confidence threshold can be chosen depending on the desired results and can be dynamically changed. The peak F1 score is 0.92. B) Census Income. The peak F1 score is 0.853 C) Breast Cancer. The peak F1 score is 0.997. D) Breast Cancer repeated but using the circuit model rather than the functional model. The peak F1 score and the shape of the curves are similar to functional model results. E) MNIST. The peak F1 score is 0.98–.99, depending on the resolution of the spike encoding. F) The individual F1 classification scores of the hand written digits.
Figure 17.
Semi-supervised operation of the AHaH classifier.
For the first 30% of samples from the Reuters-21578 data set, the AHaH classifier was operated in supervised mode followed by operation in unsupervised mode for the remaining samples. A confidence threshold of 1.0 was set for unsupervised application of a learn signal. The F1 score for the top ten most frequently occurring labels in the Reuters-21578 data set were tracked. These results show that the AHaH classifier is capable of continuously improving its performance without supervised feedback.
Figure 18.
Complex signal prediction with the AHaH classifier.
By posing prediction as a multi-label classification problem, the AHaH classifier can learn complex temporal waveforms and make extended predictions via recursion. Here, the temporal signal (dots) is a summation of five sinusoidal signals with randomly chosen amplitudes, periods, and phases. The classifier is trained for 10,000 time steps (last 100 steps shown, dotted line) and then tested for 300 time steps (solid line).
Figure 19.
Unsupervised robotic arm challenge.
The average total joint actuation required for the robot arm to capture the target remains constant as the number of arm joints increases for actuation using the AHaH motor controller. For random actuation, the required actuation grows exponentially.
Figure 20.
64-city traveling salesman experiment.
By using single-input AHaH nodes as nodes in a routing tree to perform a strike search, combinatorial optimization problems such as the traveling salesman problem can be solved. Adjusting the learning rate can control the speed and quality of the solution. A) The distance between the 64 cities versus the convergences time for the AHaH-based and random-based strike search. B) Lower learning rates lead to better solutions. C) Higher learning rates decrease convergence time.
Table 7.
Maximum power and corresponding synaptic weights.
Table 8.
Application spike sparsity and AHaH node count.