Neurogrid: A Mixed-Analog-Digital Multichip System for Large-Scale Neural Simulations

Neurogrid: A Mixed-Analog-Digital Multichip System for Large-Scale Neural Simulations
INVITED
PAPER
Neurogrid: A
Mixed-Analog-Digital Multichip
System for Large-Scale
Neural Simulations
This paper describes the design of the first hardware system to provide computational
neuroscientists with the capability of performing biological real-time simulations
of a million neurons and their synaptic connections.
By Ben Varkey Benjamin, Peiran Gao, Emmett McQuinn, Swadesh Choudhary,
Anand R. Chandrasekaran, Jean-Marie Bussat, Member IEEE , Rodrigo Alvarez-Icaza,
John V. Arthur, Paul A. Merolla, and Kwabena Boahen, Senior Member IEEE
ABSTRACT | In this paper, we describe the design of Neurogrid,
a neuromorphic system for simulating large-scale neural
3) we interconnected neural arrays in a tree network; this choice
maximized throughput. These three choices made it possible to
models in real time. Neuromorphic systems realize the function
simulate a million neurons with billions of synaptic connections in
of biological neural systems by emulating their structure.
real timeVfor the first timeVusing 16 Neurocores integrated on a
Designers of such systems face three major design choices:
board that consumes three watts.
1) whether to emulate the four neural elementsVaxonal arbor,
|
Analog circuits; application specific integrated
synapse, dendritic tree, and somaVwith dedicated or shared
KEYWORDS
electronic circuits; 2) whether to implement these electronic
circuits; asynchronous circuits; brain modeling; computational
circuits in an analog or digital manner; and 3) whether to
interconnect arrays of these silicon neurons with a mesh or a
neuroscience; interconnection networks; mixed analog-digital
integrated circuits; neural network hardware; neuromorphic
tree network. The choices we made were: 1) we emulated all
electronic systems
neural elements except the soma with shared electronic circuits;
this choice maximized the number of synaptic connections; 2) we
realized all electronic circuits except those for axonal arbors in an
analog manner; this choice maximized energy efficiency; and
Manuscript received November 27, 2013; revised February 9, 2014; accepted
February 25, 2014. Date of publication April 24, 2014; date of current version April 28,
2014. This work was supported by the National Institutes of Health Director’s Pioneer
Award DPI-OD000965 and Transformative Research Award R01NS076460.
B. V. Benjamin is with the Department of Electrical Engineering, Stanford University,
Stanford, CA 94305 USA.
P. Gao and K. Boahen are with the Department of Bioengineering, Stanford University,
Stanford, CA 94305 USA (e-mail: [email protected]).
E. McQuinn is with In-Depth, Inc., San Francisco, CA 94107 USA.
S. Choudhary is with Intel Corporation, Santa Clara, CA 95054-1537 USA.
A. R. Chandrasekaran is with Mad Street Den, Bangalore 782003, India.
J.-M. Bussat was with the Department of Bioengineering, Stanford University,
Stanford, CA 94305 USA.
R. Alvarez-Icaza, J. V. Arthur, and P. A. Merolla are with the Almaden Research
Center, IBM, San Jose, CA 95120 USA.
Digital Object Identifier: 10.1109/JPROC.2014.2313565
I. SIMULATING LARGE-SCAL E
NE URAL MODELS
Large-scale neural models seek to integrate experimental
findings across multiple levels of investigation in order to
explain how intelligent behavior arises from bioelectrical
processes at spatial and temporal scales six orders of
magnitude smaller (from nanometers to millimeters and
from microseconds to seconds). Due to prohibitively
expensive computing costs, very few models bridge this
gap, failing to make behaviorally relevant predictions [1]. A
personal computer simulates a mouse-scale cortex model
(2.5 106 neurons) 9000 times slower than a real mouse
brain operates [2], while using 40 000 times more power
(400 W versus 10 mW [3]). Simulating a human-scale cortex
model (2 1010 neurons), the Human Brain Project’s goal, is
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Vol. 102, No. 5, May 2014 | Proceedings of the IEEE
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projected to require an exascale supercomputer (1018 flops)
[4] and as much power as a quarter-million households
(0.5 GW) [5]. Hence, large-scale neural modeling’s potential
has hardly been tapped.
Several groups are developing custom computing
platforms with the stated aim of simulating large-scale
neural models affordably. The University of Manchester
SpiNNaker project aims to improve the performance of
software simulations by integrating 18 mobile processors
onto a single die [6], [7]. The IBM SyNAPSE project
(GoldenGate chip) aims to overcome the memory bottleneck such simulations face by replacing virtualization with
custom-designed digital electronic circuits that are each
dedicated to emulating a single neural element [8]. The
Heidelberg University BrainScales project (HICANN chip)
aims to reduce the number of transistors these electronic
circuits require by using an analog approach [9]. Our
Neurogrid project (Fig. 1), which also uses an analog
approach, aims to reduce transistor count further by
sharing synapse and dendritic tree circuits [10]. Thus, these
four projects have adopted radically different architectures.
They also use different interconnection networks to route
spikes between arrays of neural elements.
In this paper, we present an analysis of various
neuromorphic architectures’ and spike-routing networks’
scaling properties, which informed Neurogrid’s design
choices (Section II); describe the complete Neurogrid
system (Section III); provide detailed descriptions of
Neurogrid’s neuron circuit (Section IV) and chip design
(Section V); dissect Neurogrid’s energy consumption
(Section VI); compare its area, energy, and time per
synapse or synaptic activation with the other systems
under development (Section VII); and discuss insights
gleaned from these comparisons (Section VIII).
II. NEUROMORPHIC ARCHITECTURES
Neuromorphic hardware [11], usually realized with axonal
arbor, synapse, dendritic tree, and soma elements, may be
Fig. 1. Neurogrid. (a) GUI: Enables a user to change his or her model
parameters (left), view spike activity in the model’s various layers
(middle), plot spike rasters from a selected neural layer (right), and
enter commands (bottom). (b) Board: Each neural layer is simulated by
up to 256 256 silicon neurons on each of 16 Neurocores integrated on
a 6.5 7.5 in2 board.
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Fig. 2. Analog and digital silicon neurons. (a) Analog implementation:
Incoming spikes on the vertical wire (axon) meter charge (synapse)
onto the horizontal wire (dendrite), whose capacitance integrates the
charge. The comparator (soma) compares the resulting voltage with a
threshold and triggers an outgoing spike when the threshold is
exceeded. The capacitor is then discharged (reset) and the cycle starts
over. (b) Digital implementation: A counter is incremented (dendrite)
each time a 1 is read out of a bit cell (synapse), triggered by the
incoming spike (axon). The counter’s output is compared (soma) with a
digitally stored threshold and a spike is triggered when it is
suprathreshold. The counter is then reset and the cycle starts over.
categorized by architecture (i.e., whether elements are
dedicated or shared) and implementation (i.e., whether
elements are analog or digital). These distinctions yield
many neuromorphic hardware realizations. In this section,
we briefly review four neural-element array architectures
and five hardware realizations, as well as two interconnection network topologies for routing spikes. In addition
to bringing readers unfamiliar with the field up to speed,
this review motivates the architectural and implementation choices we made in designing Neurogrid. Before
proceeding, we describe simple analog and digital implementations of the four neural elements.
Axonal arbors, synapses, dendritic trees, and somas
may be implemented in an analog or digital fashion: In the
simplest fully analog implementation, these elements are
emulated by a wire, a switched current-source, another
wire, and a comparator, respectively [12] [Fig. 2(a)]. The
switched current source’s bias voltageVwhich determines
the synaptic weightVis stored in an analog [13] or digital
[14] manner; the latter requires a digital-to-analog
converter. In the simplest fully digital implementation,
the switched current source is replaced with a bit cell, the
axon and dendrite function as word and bit lines,
respectively, and integration and comparison are implemented digitally [8], [15] [Fig. 2(b)].
A. Four Architectures
We review four distinct architectures: fully dedicated
(FD), shared axon (SA), shared synapse (SS), and shared
dendrite (SD).1 In FD, first realized in very large scale
integration (VLSI) 29 years ago [17], [18], all elements are
dedicated [Fig. 3(a)]. A fully connected network of N
1
Shared-soma architectures have been reviewed extensively elsewhere [16].
Benjamin et al.: Neurogrid
neurons requires N 2 synapse elementsVa feature this
architecture shares with SA.
In SA, first realized 22 years ago using the address–
event representation (AER) [19], [20], each neuron is
assigned a unique address. Each time any neuron spikes, its
address is encoded by a transmitter, communicated on a
digital bus, and decoded by a receiver [Fig. 3(b)]. This timemultiplexing leverages digital’s speed to reduce the number
of wires from N to log2 ðNÞ without any noticeable penalty
as long as the bus does not become overly congested [21].
In SS, first realized 24 years ago [22], [23], only N
electronic circuits are required to fully connect N neurons,
rather than N 2 , a feature shared with SD [Fig. 3(c)]. SS
uses RAM to realize axonal branchingVinstead of
connecting a dedicated wire to multiple synapse circuits
as FD and SA do. In the original realization, which was all
digital, each shared-synapse circuit retrieved the weight to
be applied from a local RAM using the address on the AER
bus (unlike in the figure) [22], [23]. As it limits an
individual synapse’s effect to its allotted time slot, this
digital realization is said to be time-multiplexed. SS is
attractive when connectivity is sparseVa situation in
which FD and SA waste hardware implementing weights
that are zero.
More recent shared-synapse designs have used an
analog realization and consolidated all the RAM in a single
monolithic block [24]–[26], which may be embedded in
the chip [27]–[29]. The target neurons’ addresses and
weights are written to a location specified by the source
neuron’s address and retrieved sequentially. The analog
realization allows an individual synapse’s effect to extend
beyond its allotted time slot, decaying exponentially with
time. Because this behavior is realized via the principle
of linear superposition, such shared-synapse circuits are
said to be superposable [30]–[32], as opposed to timemultiplexed. A resistor–capacitor circuit (which may be
implemented with transistors [33]) is used; it essentially
performs temporal low-pass filtering.
Fig. 3. Functionally equivalent architectures. (a) Fully dedicated:
Hardware elements (for axonal arbors, synapses, dendritic trees, and
somas) are dedicated to individual neural elements. Thus, there is a
one-to-one correspondence between the chip’s elements and the
neural network’s elements. (b) Shared axon: A common set of wires is
shared by all of a neuronal population’s axons. (c) Shared synapse:
A single electronic circuit is shared by all of a neuron’s synapses; a RAM
is programmed to route all of its presynaptic spikes to this circuit.
(d) Shared dendrite: A single resistive network is shared by a neuronal
population’s (overlapping) dendritic trees.
Table 1 Architecture Comparison
In SD, first realized 10 years ago [34], [35], each
shared-synapse circuit feeds its neuron’s neighbors as well.
This arrangement models a cluster of synapses formed by
an axon onto dendrtitic branches from nearby neurons
(i.e., overlapping dendritic trees). The shared-dendrite
circuit makes this possible by applying the superposition
principle in space instead of in time. In its analog
implementation, a resistive network (which may also be
implemented with transistors [36]) is used; it essentially
performs spatial low-pass filtering.
B. Realization Comparison
We compare a total of five realizations: fully dedicated
analog (FDA), shared axon hybrid (SAH), shared axon
digital (SAD), shared synapse hybrid (SSH), and shared
dendrite hybrid (SDH). The hybrid realizations are all
analog except for their axonal arbors. To compare these
five realizations, we calculated how the area ðAÞ a single
synapse occupies, the energy ðEÞ consumed when it is
activated, and the time ðTÞ it takes to do so scale with the
number of neurons ðNÞ in a fully connected network
(Table 1). We ignore other important metrics, in particular
precision. This is intentional because neuromorphic
systems seek to compute with low-precision elements;
their raison d’être is to achieve precision at the network
level by leveraging collective computation.2
A is equal to Aarray =N 2 , where Aarray is the array’s area,
which is N 2 units for FDA, SAH, and SAD, and N units for
SSH and SDH.
E is equal to ðEaxon þ naxon Edend Þ=naxon , where Eaxon
and Edend are, respectively, the energy required to activate
an axon and a dendrite, obtained by multiplying the wire’s
capacitance (proportional to its length) by its voltage
swing; naxon is the number of synapses the axon contacts.
For the axon, the voltage swing is 1 unit (normalized by the
supply voltage VDD); hence, Eaxon is N units for FDA, SAH,
2
The relationship among precision, power, and area is well understood at the element level; it favors analog over digital at low precision
(G 8 b [37]).
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Table 2 RAM Costs
p
and SAD, and 2 N units for SSH and SDH. For the
dendrite, the voltage swing is 1 unit for digital (full swing)
and 1=N units for analog (1=N th full swing).3 Hence, Edend
is N units for SAD, 1 unit for FDA and SAH, and 1=N units
for SSH and
p SDH. naxon is N for FDA, SAH, and SAD, 1 for
SSH, and N for SDH.
T is equal to taxon =ðnpar naxon Þ, where taxon is the time it
takes to activate an axon (proportional to its capacitance)
and npar is the number of axons that can be activated in
parallel (1 in all cases but FDA, where it is N). Except for
SAD, where we must add the time it takes to activate a
dendrite (tdend ¼ N units) divided by the number of
dendrites activated in parallelp
ðnaxon ¼ NÞ. taxon is N units
for FDA, SAH, and SAD and N units for SSH and SDH.
Note that full connectivity requires T ¼ 1=N 2 (i.e., N 2
bandwidth), which none of these realizations achieve.
We also have to account for the cost of SSH’s and
SDH’s RAM, whose size equals the number of addressable
locations times the number of words per location (Table 2).
SSH requires an N N RAM, whose A, E, and T scale p
like
SAD’s synapse array’s. SDH only requires an N N
RAM,
p as the shared dendrite provides
p an additional fanout
of N. As a result, A and E are N times smaller, and
hence SDH’s AET
p product scales like
p SAH’s.
p Partitioning
this RAM into N banks (of size
N
N) reduces E
p
and T by an additional factor of N, making SDH’s AET
product scale like FDA’s.
When the cost metric is the product of A, E, and T, FDA
and SDH tie for the lowest cost and SAD has the highest,
N2 times more costly. Giving A, E, and T the same
exponents favors achieving performance through parallelism rather than by burning power.4 Interestingly, FDA
achieves its cost-effectiveness by minimizing T, while SDH
does it by minimizing A (see Table 1). That is, for an
N-neuron network, the former runs N times faster but the
latter uses N times less areaVand constrains connectivity
3
It makes the neuron’s spike rate similar to the N neurons synapsing
onto it.
4
T can be halved by giving each neuron two shared-synapse circuits
that can be activated concurrently, which doubles A; or by doubling the
voltage, which quadruples E [38].
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patterns. Thus, FDA is the best choice for applications that
run faster than real time and have arbitrary connectivity,
such as modeling neural development (HICANN’s goal
[9]), while SDH is the best choice for applications that
require lots of neurons but have mostly local connectivity,
such as modeling neocortex (Neurogrid’s goal [10]).
Our conclusions do not change when static power
dissipation (due to bias or leakage currents) is included in
the model. FDA and SDH use complementary strategies to
reduce static dissipation. FDA runs its N 2 physical
synapses N times faster, reflected in its N-fold lower T,
decreasing static energy proportionally. SDH realizes its
N 2 synapses with N times fewer transistors, reflected in its
N-fold lower A, decreasing static power proportionally.
Since both have AT ¼ 1, static energy makes similar
contributions to E in both. This result predicts that FDA
and SDH have similar static power (per synapse) and
suggests how to extend the model to include it.
C. Spike-Routing Networks
Meshes [6], [39] and trees [40]–[42] have been
explored for routing spikes between neural-element arrays
(called nodes), with a route appended to the neuron’s
address to create a packet. We compare how these
networks’ throughput and latency scale with the number
of nodes n.5 Because each neuron is connected to
thousands of others, high bandwidth is required. And
because spike times are used to encode information,
extremely short latency is required.
Meshespoffer high bandwidth due to their large channel
bisection ( n for n nodes),
p but have long latency due to
their large diameter ð nÞ.6 Trees offer short latency
(diameter is log n) but have low bandwidth (channel
bisection is 1). Unlike meshes, however, trees support
deadlock-free multicast communication (i.e., routing one
packet to many destinations), enabling them to utilize
their limited bandwidth efficiently, and thereby maximize
throughput.
Deadlock occurs when a packet is waiting for a packet
ahead of it to move, which in turn is waiting for a packet
ahead of it, and so on, and these dependencies form a
closed cycle. In this case, none of the packets make
progress toward their destinations. Hence, the routing
network is said to be deadlocked. For unicast communication (i.e., one-to-one routing), meshes are provably
deadlock-free when dimension-order routing or virtual
channels are used [43]. However, this solution does not
work for multicast routing, which introduces additional
5
We ignore all terms in the expressions except the one with the
largest exponent, and we drop this term’s coefficient (e.g.,
ð3=2Þn2 þ 8n ! n2 ).
6
Channel bisection is defined as the minimum number of links
connecting two halves of the network across all possible bisections [43].
Diameter is defined as the longest minimal path, in number of links
transversed, between any pair of nodes [43].
Benjamin et al.: Neurogrid
Table 3 Mesh Versus Tree for All-to-All Traffic
packet dependencies [44]. Trees are provably deadlock-free
in the unicast case when up-down routing is used [45]–
[47]. This solution works for the multicast case as well if
branching (copying a packet from a parent to its two
children) is restricted to the downward phase [42].
The number of packets each node in a mesh or a tree
relays can be obtained analytically for all-to-all traffic
(Table 3) [42], [47]. In this extreme benchmark, n nodes
exchange a total of n2 packets. For the mesh, which is
restricted to unicast to avoid deadlock, each node relays
n3=2 packets (traffic is uniformly distributed). For the tree,
the root relays n2 packets for unicast and n for multicast.
Thus, multicast cuts the root’s traffic by a factor of n (equal
to the fanout), relieving this
p bottleneck. As a result, the
tree’s peakpnode traffic is n less than the mesh’s, and its
latency is n= logðnÞ times shorter.
To summarize, in addition to its lower latency, the tree
offers higher throughput than the mesh if the application
can utilize the former’s multicast capability, and it
requires roughly a third less resources than the mesh
does.7 As the neocortical simulations Neurogrid targets
can use multicast to realize secondary axon-branching, we
chose the tree.
The routing network may be incorporated into the
scaling model developed in Section II-B by including its
area in calculating A, its energy in calculating E, and its
bandwidth in calculating T (Vainbrand and Ginosar [48]
analyze how these quantities scale for various topologies).
We illustrate this for Neurogrid in Sections VI and VII.
II I. NEUROGRID
Neurogrid has two main components: software to perform
interactive visualization and hardware to perform realtime simulation (Fig. 4). Neurogrid’s software stack is
composed of a user interface (UI), a hardware abstraction
layer (HAL), and driver components (Driver). UI allows a
user to specify models of neural networks to be simulated,
interact with the simulations, and visualize the results in
real time. HAL maps the parsed model description to
Neurogrid’s electronic circuits. Driver programs this
mapping on to Neurocores over USB using Neurogrid
packets.
7
The hardware grows quadratically with the number of connections a
node has, which, including the connection to its array, is four for a binary
tree and five for a 2-D mesh.
Fig. 4. Neurogrid’s software and hardware. UI: NGPython allows a user
to specify neuronal models in the Python programming environment;
GUI provides an interface to control simulations as well as to
visualize the results in real time. HAL: Network and Filter
Manager provide the GUI with the simulated network’s
connectivity and activity, respectively; Platform Control converts
the user’s neural models’ parameters to bias currents for Neurogrid’s
electronic circuits; Experiment Control starts, stops, and resets both
simulation and stimulation; Network Mapping converts the
models’ connectivity to router configurations; Data Flow translates
data from model space to hardware space and vice versa. Driver:
Neurogrid Control handles global resets and bring-up; Neurocore
Configuration programs bias currents and router configurations; Data
Flow creates Neurogrid packets; Encoder converts Neurogrid packets
to USB format and Decoder converts USB data to Neurogrid
packets. Board: FX2 handles USB communication with the host; CPLD
converts USB data to Neurogrid packets and vice versa, as well as
intersperses time stamps with outgoing data (host bound);
Daughterboard realizes primary axonal branching. Neurocore: Router
communicates packets with the Neurocore’s parent (through T i =T o )
and two children (through Li =Lo and R i =R o ); RAM supports
reconfigurable connectivity (a second RAM supports programmable
biases); Receiver delivers spikes to silicon neuron array; Transmitter
dispatches spikes from the array. Neuron: Consists of a soma, a
dendrite, four gating variable and four synapse-population circuits.
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I V. NEURON
Neurogrid neuron’s soma, dendrite, synapse-population
and ion-channel-population circuits realize the dimensionless form of common biological neuronal models with
MOS devices. Dimensionless models have fewer free
parameters and can be realized on a wide variety of
hardware platforms. However, MOS devices offer the
highest integration density.
A. Dimensionless Models
Before describing the dimensionless models Neurogrid
realizes (Fig. 6), we illustrate how models composed of
conductors, capacitors, voltage, and current sources may
be converted to dimensionless form.
Consider a passive membrane model with a capacitor C,
a conductor Gleak with reversal potential Eleak , and a
current source Iin . This circuit is described by
Fig. 5. Neurocore. RAM0 provides 256 locations for target synapse
types (or no connection). RAM1 stores 18 configuration bits and
61 analog biases, common to all the Neurocore’s silicon neurons.
DACs produce the analog biases. RstMB provides five resets and
generates DACs’ reference current. ADCs digitize four analog signals
from a selected neuron. T i02 , T o02 , Li02 , Lo02 , R i02 , and
Ro02 communicate with parent or either child. The 12 14 mm2 die,
with 23 M transistors and 180 pads, was fabricated in a 180-nm
complementary metal–oxide–semiconductor (CMOS) process.
Insert: Layout of silicon neuron; it has 337 transistors (see Fig. 4
for subcircuit placement).
A Neurogrid packet, used in Neurocore-to-Neurocore
communication, is a sequence of 12-b words that specify a
route, an address, an arbitrarily long payload, and a
tailword, in that order. The route instructs a Neurocore to
forward the packet to the next hop or consume it. The
payload stipulates: Spike locations in a row of a siliconneuron array, in which case the address identifies a
Neurocore; data to be written to RAM, in which case the
address specifies a location; or a sampled analog signal, in
which case the address identifies an ADC (a Neurocore has
four). The tailword signifies the packet’s end.
The hardware consists of an Cypress EZ-USB FX2LP, a
Lattice ispMACH CPLD, a daughterboard, and 16
Neurocores connected in a binary tree. The FX2 handles
USB communications. The CPLD interfaces between the
FX2 and the Neurocores. The daughterboard realizes
primary axon-branching using a Xilinx Spartan-3E FPGA
and eight Cypress 4MB SRAMs. A Neurocore (Fig. 5) has a
256 256 silicon-neuron array, a transmitter, a receiver, a
router, and two RAMs. A neuron has a soma, a dendrite,
four gating-variable and four synapse-population (i.e.,
shared synapse and dendrite) circuits. We describe these
electronic circuits and the models they implement in
Section IV. The transmitter, the receiver, and the router are
described in Section V.
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CV_ ¼ Gleak ðV Eleak Þ þ Iin
where V is the voltage across C. This equation has four
parameters: C, Gleak , Eleak , and Iin , even though the model has
only two degrees of freedom. To see this, change the
reference voltage to Eleak and normalize with Gleak Vn to give
v_ ¼ v þ u
(1)
Fig. 6. Neurogrid’s neuron model. Synapse Populations 0, 1, 2, and 3
generate a time-varying conductance ðgsyn03 Þ in response to an input
spike and drive neighboring neurons’ soma or dendrite through a
shared dendritic tree. Channel Populations 0 and 1 provide a pair of
conductances that are dynamically activated or inactivated ðc03 Þ by
dendritic potential; their maximum conductance may be determined
by Synapse Population. These conductances may be connected in
series or in parallel to drive Dendrite. Dendrite also receives
backpropagating spikes ðibp Þ and drives Soma. Soma generates spikes,
using a regenerative sodium current ðiNa Þ, which triggers a reset
pulse ðpres Þ. A potassium conductance ðgK Þ activated at reset delays
generation of the next spike.
Benjamin et al.: Neurogrid
Fig. 8. Dendrite circuit’s membrane traces. (a) Increasing d : It
Fig. 7. Soma circuit’s membrane traces. (a) Increasing s : It increases
the interspike interval by slowing integration. (b) Increasing tres :
It increases the interspike interval by resetting the membrane longer.
(c) Increasing gK1 : It increases the interspike interval by producing
larger increments in the potassium conductance.
increases decay time by slowing integration (traces were normalized
with their peak values). (b) Increasing ibp : It increases the current
injected by each backpropagating spike.
Synapse Population: The synapse population’s dimensionless model is given by
where ¼ C=Gleak , v ¼ ðV Eleak Þ=Vn , and u ¼ Iin =
ðGleak Vn Þ. Thus, v is the voltage in units of Vn and u is
the current in units of Gleak Vn . This equation has two
parameters: and u, to match the model’s two degrees of
freedom. This approach is general: Any electrical model
of a membrane can be made dimensionless by changing
the reference voltage to Eleak and normalizing voltages
with Vn , conductances with Gleak and currents with
Gleak Vn . Henceforth, we denote dimensionless equivalents of voltages with v, conductances with g, and
currents with i.
(2)
where s is the membrane time constant, isin is the input
current, and vd is the dendritic input (see Fig. 6 Soma).
The quadratic positive feedback v2s =2 models the spikegenerating sodium current [49]; the reset conductance gres ,
active for the duration tres of a unit-amplitude pulse pres ,
models the refractory period; and the high-threshold
potassium conductance gK models spike-frequency adaptation. gK is given by
K g_K ¼ gK þ gK1 pres ðtÞ
(5)
where syn is the synaptic time constant and gsat is the
saturation conductance for the population (see Fig. 6
Synapse Population). The unit-amplitude pulse prise ðtÞ is
triggered by an input spike; its width trise models the
duration for which neurotransmitter is available in the
cleft [50]. Hardware realization of the synapse-population
model behaves as expected [Fig. 9(a)–(d)].
The conductance gsyn decays spatially in the shared
dendritic tree and provides an input current
ðnÞgsyn ðesyn vs Þ
Soma: The soma’s dimensionless model is given by
s v_s ¼ vs þ isin þ 21v2s gK vs gres vs pres ðtÞ þ vd
syn g_syn ¼ gsyn þ gsat prise ðtÞ
or
ðnÞgsyn ðesyn vd Þ
(6)
to the soma or dendrite, respectively (in addition to isin or
idin ). Here
41 !2 n
1
1
p
1þ
(7)
ðnÞ ¼ p
4 1 2
n
(3)
where K is the decay time constant and gK1 is the
saturation value. The soma may also receive synaptic
inputs [see (6)]. Hardware realization of the soma model
behaves as expected (Fig. 7).
Dendrite: The dendrite’s dimensionless model is
given by
d v_d ¼ vd þ idin þ ibp pres ðtÞ þ gch ðech vd Þ
(4)
Fig. 9. Synapse population circuit’s conductance traces. (a) Increasing
where d is the membrane time constant, idin is the input
current, ibp is the backpropagating input, and gch is the
channel population’s conductance, with reversal potential
ech (see Fig. 6 Dendrite). The dendrite may also receive
synaptic inputs [see (6)]. Hardware realization of the
dendrite model behaves as expected (Fig. 8).
trise : It prolongs the rising phase. To keep the area constant, gsat was
divided by trise . (b) Increasing gsat : It increases the synaptic conductance
proportionately. (c) Increasing syn : It slows integration, resulting in
smaller peak conductances and longer decay times. (d) Increasing esyn :
It changes the effect on the membrane potential from inhibitory (red)
to excitatory (black). (e) Increasing (left to right, top to bottom):
It increases the spread of synaptic conductances evoked at six
locations, arranged in a hexagon, by a spike delivered to their center.
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Benjamin et al.: Neurogrid
where is the silicon dendritic tree’s decay factor and n is
the distance traveled in a number of neurons [51].
Hardware realization of the dendritic tree behaves as
expected [Fig. 9(e)].
Ion-Channel Population: The ion-channel population’s
conductance gch is obtained by scaling a maximum
conductance gmax with a gating variable c (i.e.,
gch ¼ cgmax ; see Fig. 6 Channel Population). c is modeled
as
gv c_ ¼ c þ css
(8)
where css is its steady-state activation or inactivation and
ch is its time constant. css is given by
css ¼
þ
or
þ
(9)
where and model a channel’s opening and closing
rates, whose voltage dependence is modeled as
rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
1
1
1
; ¼ ðvd vth Þ þ
ðvd vth Þ2 þ 2 :
(10)
2
2
4s
Here, vth is the membrane potential at which css ¼ 1=2
and s is the slope at this point. and satisfy a difference
relation ¼ vd vth and a reciprocal relation ¼
1=ð16s2 Þ, resulting in a sigmoidal dependence of css on vd .
The gating variable’s time constant is given by
gv ¼
max min
þ min :
2sð þ Þ
Fig. 10. Gating-variable curves from ion-channel population circuit.
(a) Steady-state value increases (for activation) with increasing
membrane potential, exhibiting a sigmoidal dependence. Increasing
s increases the slope. (b) Time constant has a bell-curved dependence
on membrane potential. max scales the peak. (c) Threshold increases
with increasing vth .
(11)
gv is bell shaped with a maximum value of max when
vd ¼ vth and a minimum value of min when jvd vth j 1=ð2sÞ, to avoid unphysiologically short time constants.
Hardware realization of the ion-channel population model
behaves as expected (Fig. 10).
B. Circuit Realization of Dimensionless Models
Before describing the dimensionless models’ circuit
implementations, we illustrate how such models may be
realized in the log domain [33] by using MOS devices
operating in the subthreshold regime to realize a passive
membrane [see (1)].
In the subthreshold regime, a PMOS transistor’s drain
current Id is related to its gate-bulk voltage Vgb by
V V Vds gb sb
Id ¼ LI0 e UT
1 e UT
(12)
source voltages, respectively [38]. For Vds G 4UT and
Vsb ¼ 0, (12) is approximately
Id ¼ LI0 e
Vgb
UT
:
Taking natural logarithm on both sides
ln Id ln I0 ln L ¼ Vgb
:
UT
(13)
Differentiating (13) with respect to time, we obtain
I_d
¼ V_ gb :
UT
Id
(14)
These equations are the basis for realizing models in the
log domain with MOS transistors.
The passive membrane’s circuit realization (Fig. 11)
consists of a capacitor ðCÞ, with the voltage across it ðVm Þ
driving a transistor ðM5 Þ to produce an output current ðIm Þ
that represents the membrane’s potential. A current
sourced into the capacitor ðIlk Þ represents the membrane’s
leak and a current sunk from the capacitor ðIback Þ represents
the membrane’s input. Kirchoff’s current law gives
CV_ m ¼ Ilk Iback :
(15)
where L ¼ W=L is the transistor’s width-to-length ratio, I0
is the leakage current when L ¼ 1, is the ratio between
effective and applied gate voltage, and UT is the thermal
voltage8; and Vsb and Vds are the source-bulk and drainFig. 11. Passive membrane circuit. It models the membrane’s time
8
UT ¼ kT=q, where k is the Boltzmann constant, T is the absolute
temperature, and q is an electron’s charge.
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Proceedings of the IEEE | Vol. 102, No. 5, May 2014
constant (through Ileak ), input current u (through Iin ), and
potential v (through Im ). Bulk terminals are connected to the power rail
unless otherwise indicated.
Benjamin et al.: Neurogrid
Table 4 Mapping Constants
Fig. 12. Soma circuit. MEM models membrane time constant s
(through Ilks ), input current isin (through Ibks ), and dendritic
input vd (through Id ; see dendrite circuit). QF models quadratic
feedback vs2 =2 (through Ian ). Kþ models high-threshold potassium
conductance (through Ilkk and Ik1 ). Ref models reset conductance gres
and refractory pulse pres (through Ilkref ).
As M1 ’s and M3 ’s gate-bulk voltages are equal, we have
Vleak ¼ Vlk )
Ileak Ilk
¼
:
L1
L3
(16)
As the sum of M1 ’s and M2 ’s gate-bulk voltages is equal to
the sum of M4 ’s and M5 ’s, we have
Vleak þ Vin ¼ Vback þ Vm )
Ileak Iin Iback Im
¼
:
L1 L2
L4 L5
which is equivalent to (2). Here
Ins ¼
v
v_
(17)
(18)
where p ¼ CUT L1 =L3 and pin ¼ L4 L5 =L2 L3 are the
mapping constants required to program Ileak and Iin to
realize the desired values of and u.9
Soma: The soma model’s circuit realization (Fig. 12)
operates according to
v_s
isin
I2lks
;
kd I0
and
tres ¼
pref
Ilkref
g_K
gK
gK1
K
z}|{ zfflfflfflffl}|fflfflfflffl{
z}|{ z}|{
pK I_K
IK
IK1
¼
þ pgK
pres ðtÞ
Ilkk InK
InK
InK
u
v2s
2
Ind ¼
and IVDD is the drain current of a PMOS with gate
grounded and source at the supply rail. All mapping
constants are defined in Table 4 (in this table, Ln is the
W=L of transistor Mn ). The high-threshold potassium
conductance’s circuit realization operates according to
Using (14), (16), and (17) in (15) yields
CUT I_m
L3
L4 L5 Ileak Iin
¼ Ileak þ
Im
L1
L1 L2 Im
_
p Im
Im
Iin
)
¼
þ pin
Ileak Ileak
Ileak
Ileak
|{z} |{z}
|{z} |fflfflffl{zfflfflffl}
Ilks
;
ks
which is equivalent to (3). Here, InK ¼ Ilks =K . Detailed
soma-circuit descriptions may be found elsewhere [54], [55].
Dendrite: The dendrite model’s circuit realization
(Fig. 13) operates according to
v_
ibp
idin
d
vd
d
ffl{
z}|{ zfflfflfflfflfflfflffl}|fflfflfflfflfflfflffl{ zfflfflfflfflfflffl}|fflfflfflfflffl
z}|{ z}|{
I2bp
pd I_d
Id
I2bkd
¼
þ pbkd
þ pbp
pres ðtÞ
Ilkd Ind
Ind
Ind Ilkd
Ind Ilkd
which is equivalent to (4).
g
vs
vs
K
s
z}|{ zfflfflfflffl}|fflfflfflffl{ z}|{ zfflffl}|fflffl{ z}|{
z}|{ z}|{
2
2
_
ps Is
Is
I
1I
IK Is
¼
þ pqua bks
þ s K
Ilks Ins
Ins
I2lks 2 I2ns
Ilks Ins
IVDD Is
Id
res
pres ðtÞ þ
Ilks Ins
Ind
|fflfflfflfflffl{zfflfflfflffl
ffl} |{z}
|{z}
gres
vs
vd
9
To make independent of temperature, we used a proportional-toabsolute-temperature (PTAT) current source to generate Ileak [52], [53],
which keeps UT =Ileak constant.
Fig. 13. Dendrite circuit. MEM models membrane time constant d
(through Ilkd ) and input current idin (through Ibkd ). BP models
backpropagating input ibp (through Ibp ).
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707
Benjamin et al.: Neurogrid
Synapse Population: The synapse population model’s
circuit realization (Fig. 14) operates according to
g_ gsyn
gsyn
gsat
syn
zfflffl}|fflffl{ zfflfflfflfflfflfflffl}|fflfflfflfflfflfflffl{
zffl}|ffl{ zfflffl}|fflffl{
psyn I_gsyn
Igsyn
Igsat
¼
þ pgsat
prise ðtÞ
Ilklpf Ingsyns
Ingsyns
Ingsyns
where trise ¼ pc =Ilkpe and Ingsyns ¼ Ilks =gsyns .
When fed to the soma or dendrite circuit through the
dendritic-tree and reversal-potential subcircuits, Igsyn yields
the (normalized) synaptic input to soma or dendrite as
gsyni
esyni
vs;d
zfflfflffl}|fflfflffl{ zfflfflfflfflffl}|fflfflfflfflffl{ z}|{ Igsyn
Iesyn Is;d
pesyn
Ingsyns;d
Ins;d Ins;d
which is equivalent to (6). Here, Ingsynd ¼ Ilkd =gsynd . A
detailed synapse-circuit description may be found elsewhere [32].
The dendritic-tree circuit spreads the current Igsyn
through a hexagonal resistive network implemented with
transistors [36]. Its decay factor is related to the voltages
Vr and Vg as
2
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi :
¼1
1þ
1 þ 8e
ðVr Vg Þ
UT
Ion-Channel Population: The ion-channel population
model’s circuit realization (Fig. 15) computes the channel’s conductance ðgch Þ directly instead of computing the
gating variable first (8) and then scaling the saturation
conductance ðgmax Þ by this. That is, the circuit operates as
follows:
gv
g_ch
gch
c
gmax
ss
zfflfflfflfflfflfflfflfflffl}|fflfflfflfflfflfflfflfflffl{ z}|{
z}|{ zfflfflffl}|fflfflffl{ zfflfflfflfflfflfflffl}|fflfflfflfflfflfflffl{
_
pgv Imax
Igv
Igv
I
Imax
¼
þ
pgvmax
ðI þ I ÞIshift Ingv
Ingv I þ I
Ingv
Fig. 15. Ion-channel population circuit. ON–OFF models opening and
closing rates and (through I and I , respectively). Kinetics models
peak time constant max (through Ishift ), activating/inactivating behavior of css (through I1 and I1 ), and maximum conductance gmax
(through Imax ). Add/multiply combines the output of two kinetics
subcircuits to model their equivalent series or parallel conductance
(when ½S0 ; S1 ¼ ½0; 1 or ½1; 0, respectively) and models reversal
potential ech (through Iech ).
for activating behavior (i.e., I1 ! 1 and I1 ! 0) and
min ¼ 0 (i.e., Isat ! 1). Here, Ingv ¼ Ilkd Iech =ðgch I0 Þ.
Ishift realizes max as pgv max Imax =Iq Ishift . I and I realize and as
;
vd
vth
zfflfflffl}|fflfflffl{
z}|{ zfflffl}|fflffl{ I;
1
Id
Ith
pth
2 Ind
LO5 Ind
Ind
s
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
2
Id
Ith
Iq 2
1
pth
þ 4 pslope
þ
Ind
Ind
Ind
2
|{z}
|fflffl{zfflffl}
|fflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflffl}
vd
vth
1=4s2
for I I or vice versa, which is equivalent to (10).
Unlike a previous implementation [56], this one supports
adjustable slopes for activation and inactivation through an
on–off circuit [57].
Another circuit combines Igv from a pair of circuits
ðIgv0;2 and Igv1;3 ) to obtain their conductances’ series or
parallel combinations (gch0 and gch1 ; see Fig. 6), modeled as
Ich0;1
8
I I
>
< gv0;2 gv1;3 ;
¼ Igv0;2 þ Igv1;3
>
:
Igv0;2 þ Igv1;3 ;
series
parallel
which drives the dendrite with a current
Fig. 14. Synapse-population circuit. PE models the rise time trise
(through Ilkpe ). MEM models time constant syn (through Ilklpf )
and saturation conductance gsat (through Igsat ). Cable models spatial
decay factor (through Vr ). Rev models reversal potential esyn
(through Iesyn ).
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Proceedings of the IEEE | Vol. 102, No. 5, May 2014
gch0;1
ech0;1
v
d
z}|{ zfflfflfflffl}|fflfflfflffl{ z}|{
Ich0;1
Iech
Id
pech
:
Ingv
Ind
Ind
Benjamin et al.: Neurogrid
The series combination is used to model channels that
activate and inactivate (second order), while the parallel
combination models independent channels (first order)
with a common reversal potential.
To correct errors in the mapping constants’
analytical expressions (see Table 4) due to deviations
from the transistor model (12) [58], we calibrated them
by measuring four types of circuit responses (Table 5):
1) dynamic current (e.g., the dendrite’s exponentially
decaying response to a step input, which we used to
calibrate pd ); 2) steady-state current (e.g., the ion-channel
population’s conductance for a given dendritic potential,
which we used to calibrate pth ); 3) steady-state spike rate
(e.g., the linear scaling of a neuron’s spike rate with its
time constant, which we used to calibrate ps [54]); and
4) spike-rate discontinuity (e.g., the onset of spiking when
the dimensionless input exceeds 0.5, which we used to
calibrate pqua [54]). These procedures yielded mapping
constants for individual circuits. We only used the median of
these distributions, which arise from transistor mismatch, as
all of a Neuorocore’s neurons share the same biases.
V. TRANSMITTER, RECE IVER,
AND ROUTER
A neuron’s spike is dispatched from its array by a
transmitter, communicated to its Neurocore’s parent and
two children by a router, and delivered to the recipient by
a receiver. All this digital circuitry is event drivenVonly
active when a spike occursVwith its logic synthesized
following Martin’s procedure for asynchronous circuits
[60], [61].
A. Transmitter and Receiver
We provide brief descriptions of the transmitter’s and
receiver’s architecture and operation; detailed descriptions
may be found in [59] and [62]–[64]. The transmitter
dispatches multiple spikes from a row and the receiver
delivers multiple spikes to a row, enhancing throughput
compared to designs that dispatch or deliver spikes one by
one [19]–[21]. These spikes’ common row address and
unique column addresses are communicated sequentially.
In the transmitter [Fig. 16(a)], two M-way arbiters,
built with M 1 two-way arbiters connected in a binary
tree, receive requests from an array of spiking neurons
with M rows and M columns. Only log2 ðMÞ logic levels are
Table 5 Responses Used to Calibrate Mapping Constants
Fig. 16. Transmitter and receiver architecture. (a) Transmitter: An
interface (I) relays requests from spiking neurons (S) to a row arbiter
(J) and dispatches the selected row’s spikes (S) in parallel while
encoding its address (Y). Another interface (I) relays the spikes from
a latch to a column arbiter (J) and encodes the selected column’s
address (X). A sequencer (SEQ) directs latches (A) to deliver the row
address, column address(es), and a tailword (T, generated by TB) to the
output port. (b) Receiver: A sequencer (SEQ) directs two different
latches (A) to load incoming row (Y) and column (X) addresses, which
are decoded to select a row and one or more columns. These select
lines are activated simultaneously, when the tailword (T) is received,
delivering spikes to the row in parallel (S). The remaining latches
operate autonomously (B), automatically overwriting old data
after it has been read. Small discs symbolize combinational logic.
Modified from [59].
traversed to select a row or column, compared to M=2, on
average, for a scanner [65], [66]. Two encoders generate a
log2 ðMÞ-bit address for each row or column selected.
Latches enable pipelining: The next row’s spikes are
dispatched from the array while the current row’s column
addresses are being encoded and sent out. In the receiver
[Fig. 16(b)], these log2 ðMÞ-bit addresses are decoded to
select one of M rows or columns. Again, latches allow the
next packet’s addresses to be decoded while the current
one’s spikes are being delivered to the array.
Neurocore uses a 256 256 version of the transmitter
and a 2048 256 version of the receiverVits eight lines
per row select one of four shared-synapses to activate, one
of three sets of analog signals to sample, or a neuron to
disable (all in conjunction with a column line). The
transmitter takes 86 ns to transfer a row’s spikes to the
array’s periphery. Then, it takes 23 ns to encode each
spike’s column address. Therefore, if there are three or
more spikes, no time is wasted waiting for the next row’s
spikes to be transferred. Thus, pipelining and parallelism
enable the transmitter to sustain a maximum transmission
rate of 43.4 Mspike/s, or 663 spike/s per neuron [42]. The
receiver decodes an additional column address every 16 ns,
sustaining a maximum rate of 62.5 Mspike/s, or 956 spike/s
per neuron [59].10
10
This paper reports measurements from a 960 320 version of the
receiver fabricated in the same technology (180-nm CMOS).
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Benjamin et al.: Neurogrid
B. Router
We provide brief descriptions of the router’s logical and
physical design; detailed descriptions may be found in
[42]. The router’s multicast capability and the Neurocores’
embedded memory realize secondary axon-branching
(limited to corresponding locations in multiple Neurocores); primary axon-branching (to arbitrary locations in
multiple Neurocores) is realized by the daughterboard’s
FPGA and SRAM.
The router routes a packet from a source to multiple
recipients in two distinct phases: a point-to-point phase
and a branching phase (Fig. 17). During the point-to-point
phase, the router steers the packet up/down or left/right,
based on a bit in the packet’s first word. During the
branching phase, the router copies the packet to both left
and right ports. These flooding packets are delivered to the
local neural array or filtered using information retrieved
from a location in the Neurocore’s 256 16-b SRAM,
specified by the packet’s second word. This approach
achieves high throughput by distributing the packet to all
its potential recipients without performing any memory
lookups [27]. These lookups, which are the slowest
operation, proceed in parallel, in contrast to a network
where lookups decide the packet’s route [6]. If the packet
is delivered, two bits retrieved from the SRAM are
appended to its row address. These bits specify which one
of the recipient’s four shared synapses to activate [27], [42].
The router’s datapath consists of four merges and three
splits (Fig. 18). For energy efficiency, the 12-b datapath
Fig. 17. Multicast routing’s point-to-point and branching phases. In the
point-to-point phase (black), node 4’s packet is routed up to node 1,
the lowest common ancestor node 4 shares with the recipients
(nodes 3 and 6). The packet is then routed down to node 3, the
recipients’ lowest common ancestor. At each node, the route field’s
most significant bit [encodes the turn (U or D, R or L)] is shifted out and
0 is shifted in. The stop code [encodes the terminus (node 3)] is all
zeroes except the MSB (S). In the branching phase (purple), the packet
visits node 3 and all its descendantsVnode 7’s SRAM is programmed
to filter the packet. A mode bit (F) determines whether the packet
floods or targets the terminus. Reproduced from [42].
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Proceedings of the IEEE | Vol. 102, No. 5, May 2014
Fig. 18. Router datapath. Merge and Merge3 feed packets to splits:
Up steers packet to its U or D port and Down steers packets to its L or
R port. When it encounters a stop code, Up deletes the packet and
Down interrogates the mode bit. If it is clear (targeted mode), it
delivers the packet to either its M1 or M2 port (a bit in the headword
decides). If it is set (flood mode), it delivers the packet to its Lo and
R o ports as well. Reproduced from [42].
was sliced into six bit-pairs, each communicated by two
transitionsVthe second returns to zeroVon one of four
lines. This 1-in-4 code requires half as many transitions as
independent bits require (1-in-2 code). Furthermore, a
single acknowledge (or enable) line is used, instead of two
[61]. The datapath’s blocks were compact enough to be
distributed throughout the IO-pad ring (see Fig. 5). Each
port’s pads were mostly placed on its corresponding side of
the chip, which facilitated building a multichip printed
circuit board with straight connections between adjacent
chips. Two-to-one multiplexing cut the number of pads per
port from 42 to 21, organized in three groups of seven pads
(two power, four signal, and one enable). Each pad group
transmits two bits with a single transition, without
returning to zero, on one of four lines (1-change-in-4
code) [67], [68], achieving a data-rate of 364 Mb/s.
Thus, each port can handle 91 Mword/s [42].
When interconnected in a binary tree, the router
delivered 1.17 Gword/s to Neurogrid’s 16 Neurocores
with no more than 1-
s jitter along the longest path [42].
Jitter is defined as the standard deviation of intervals
between packets injected at equal intervals. These
injected packets (generated by a computer) were routed
all the way up and down the tree, ejected at a leaf, and
captured by a logic analyzer. Meanwhile, the eight
Neurocores at the tree’s leaves each generated spike
packets at a rate of 9.14 Mword/s. This traffic aggregated
at the root, which received 73.12 Mword/s in total, and
flowed to all 16 Neurocores, which received 1.17 Gword/s
in total. This delivery rate corresponds to 234 Mspike/s in
normal mode (five words per spike packet) and up to
1.17 Gspike/s in burst mode (additional column addresses
are appended to the packet).
Benjamin et al.: Neurogrid
VI. ENERGY EFFICIENCY
We measured and dissected Neurogrid’s energy consumption for a million-neuron, eight-billion-synapse real-time
simulation and found that it uses energy more efficiently
than expected from its shared-dendrite architecture.
Neurogrid was programmed to simulate a recurrent
inhibitory network with 15 layers (Fig. 19), each mapped
onto a different Neurocore. The recurrent synaptic
connections were realized by multicasting spikes from
each Neurocore to all others (including itself). That
Neurocore and its three neighbors on either side where
programmed to accept its spikes, which inhibited nearby
neurons through the shared dendrite [with ¼ 0:94; see
Fig. 9(e)]. As a result, each of the model’s 983 040
neurons received 50% of its inhibition from 7980 neurons
residing in a seven-layer-thick, 19-neuron-radius cylinder
centered around it. Such recurrent inhibitory connectivity
patterns are expected to give rise to globally synchronous
spike activity, which is what we observed. The synchronized activity was rhythmic, with a frequency of 3.7 Hz;
the neurons fired 0.42 spike/s on average.
Neurogrid consumed 2.7 W during the simulation.
Since interlayer connections were between corresponding
locations (i.e., columnar), the daughterboard was not
needed. Measurements of its power consumption from a
separate study, where we demonstrate the ability to
implement arbitrary connectivity patterns [69], revealed
that it would consume 0.4 W to route the 413 000 spike/s
that the 983 040 neurons produced, yielding a total of
3.1 W, or 941 pJ per synaptic activation (for 7980 synaptic
connections per neuron).
The energy expended to activate a silicon axon’s
synapses ðEaxon Þ may be expressed as the sum of the
Fig. 19. Simulating a million neurons. The neurons were organized into
fifteen 256 256 cell layers, arranged in a ring, so the first and last
layers are nearest neighbors. Each cell layer’s neurons inhibit
neighboring neurons in its layer as well as in three neighboring layers
to either side (the central layer’s connectivity is shown). Spike rasters
(from a tenth of each layer’s neurons) reveal global synchrony, as
expected from the network’s recurrent inhibition.
Table 6 Energy Per Spike
energy used (per spike) by the soma ðEsoma Þ, transmitter
ðExmt Þ, up route ðEu Þ, daughterboard ðERAM Þ, down route
ðEd Þ, and receiver ðErcv Þ; the last term includes the shareddendrite and synapse circuits. That is
Eaxon ¼ Esoma þ Exmt þ Eu þ ncol ðERAM þ Ed þ np Ercv Þ:
Here the axon connects to np pools of neurons (secondary
branches) in each of ncol locations (primary branches).
Therefore, the total number of synapses ðnaxon Þ it activates
is ncol np nsyn , where nsyn is the average number of
synaptic connections per pool. Hence, the energy per
synaptic activation ðE ¼ Eaxon =naxon Þ is
E¼
Esoma þ Exmt þ Eu ERAM þ Ed Ercv
þ
þ
:
ncol np nsyn
np nsyn
nsyn
(19)
Terms associated with the axon’s trunk ðEsoma ; Exmt ; Eu Þ
and primary ðERAM ; Ed Þ and secondary ðErcv Þ branches
contribute small, medium, and large amounts, respectively.
These energy terms (Table 6) were determined as follows.
Esoma equals Vjack Istatic Tsoma =Ntotal , where Vjack ¼ 3 V
is the power jack’s voltage, Istatic is its current when the
neurons are quiescent,11 Tsoma is the average interspike
interval (for which E is calculated), and Ntotal ¼ 220 is the
neuron count. Istatic only approximately captured the
analog circuitry’s current draw, which has a weak
dependence on Tsoma .
Exmt;rcv equals Vjack DIxmt;rcv =ftotal , where DIxmt is the
additional current drawn when Neurocores generate ftotal
spike/s (each one routes its spikes back to itself but its
synaptic connectivity RAM filters them) and DIrcv is the
additional current drawn when one Neurocore, to which
all the other’s spikes are routed, is reprogrammed to stop
filtering them.
Eu;d equals nu;d Elink , where nu;d is the number of links
between a Neurocore and the daughterboard and Elink is
the energy a link uses to transmit a five-word spike packet
[67].12 We set nu ¼ 3:4, the average number of links from
a Neurocore to the daughterboard, and nd ¼ 2:3, oneseventh the number of links a packet traverses when it
floods from the root (i.e., np ¼ 7).
11
The daughterboard drew 87 mA, the FX2 and CPLD drew 116 mA,
the voltage regulators (three per Neurocore) drew 74 mA, and the 16
Neurocores’ IO, digital and analog circuitry drew 25, 64, and 572 mA,
respectively. These measurements were made with an HP E3631A DC
power supply.
12
We scaled this measurement, which was made on a 3-cm-long link,
to match the average length of the interneurcore links (1.1 cm), assuming
that energy is proportional to length.
Vol. 102, No. 5, May 2014 | Proceedings of the IEEE
711
Benjamin et al.: Neurogrid
Table 7 Cost Comparison
ERAM equals Vjack DIlkup =flkup , where DIjack is the
additional current the daughterboard’s power jack draws
when it performs flkup memory lookups per second.
Substituting these measured values into (19) yields
(in nJ)
E¼
22:2 þ 2612Tsoma
9:8
9:8
þ
þ
:
np nsyn nsyn
ncol np nsyn
(20)
This model predicts E ¼ 813 pJ for the synchrony
simulation, where ncol ¼ 1, np ¼ 7, nsyn ¼ 1140, and
Tsoma ¼ 1/(0.42 spike/s). This prediction is within 14%
of the measured value of E ¼ 941 pJ. This discrepancy is
probably due to ignoring the analog circuitry’s slight
increase in power dissipation with spike rate.
VI I. AE T COST COMPARISON
Neurogrid’s overall cost ðAETÞ is lower than HICANN’s [9]
and GoldenGate’s [8] (Table 7).13 We included the router’s
costs and the entire system’s power dissipation in our A and E
calculations, unlike in Section II-B. Nonetheless, our
architecture-scaling model’s predictions held up: A was
smallest for Neurogrid (SDH), E was highest for GoldenGate
(SAD), and T was smallest for HICANN (FDA). However,
Neurogrid’s A and E were smaller than predicted because it
amortized these costs over a larger number of synaptic
connections by using multilevel axon branching. And
HICANN’s T was larger than predicted because it did not
realize FDA fullyVit multiplexes 64 neurons’ spikes onto
each shared axon. As a result, Neurogrid’s AET was 50 times
lower than HICANN’s. Before discussing these results in
detail, we describe how A, E, and T were calculated.
A was calculated as Achip =ðSNchip Þ, where Achip is the
chip’s area, Nchip is its neuron count, and S is the number
of synapses (HICANN and GoldenGate) or the number of
synaptic connections (Neurogrid) per neuron (see Table 7).
In Neurogrid’ case, we used ncol ¼ 4, np ¼ 4, and
nsyn ¼ 256 (midrange values).
E was calculated as Psys =ðfavg Nsys SÞ, where Psys is the
system’s total power at an average firing rate of favg and
Nsys is its total neuron count. To account for static
dissipation, we set favg ¼ 10 spike/s for all three systems;
actually 105 spike/s for HICANN (104 speedup). Psys was
13
For comparison with HICANN and Neurogrid, we scaled GoldenGate’s
A ¼ 16 m2 , E ¼ 1.9 nJ, and T ¼ 3.9 pS from a 0.85 V–45 nm process to a
1.8 V–180 nm process using general scaling laws [70].
712
Proceedings of the IEEE | Vol. 102, No. 5, May 2014
800 W for HICANN’s 352-chip, 48-FPGA, 180 000-neuron
wafer-scale system [71] and 5 mW for GoldenGate’s
256-neuron chip [8]. For Neurogrid, we obtained E
using (20).14
T was calculated as taxon =ðnpar naxon Þ, identical to
Section II-B. taxon was 5.21 ns for HICANN (i.e.,
192 Mspike/s), 7.56 ns for GoldenGate (scaled from a 1-ns
cycle-time estimate for a 45-nm SRAM), and 16.0 ns for
Neurogrid. We used the receiver’s word rate, since it is
slower than the router’s (13.7 ns at 1-
S jitter). npar was
224 for HICANN and 1 for GoldenGate and Neurogrid.
naxon was 8 for HICANN and 256 for GoldenGate and
Neurogrid (using shared dendrite).
To understand why Neurogrid’s AET was lower than
expected, consider the case where neither primary nor
secondary axon branching is used (i.e., np ¼ ncol ¼ 1),
which makes Neurogrid’s S similar to HICANN’s. The
resulting 16-fold drop increases A 16-fold, increases E
tenfold [using (20)] and leaves T unchanged. As a result,
Neurogrid’s AET becomes three times larger than
HICANN’s. Therefore, Neurogrid achieved lower than
expected AET by utilizing multilevel axon branching to
amortize its fixed area and static energy costs over more
synaptic connections. With S matched, Neurogrid’s E is
six times larger than HICANN’s, probably because
Neurogrid’s neuron has four shared-synapse and four
shared-dendrite circuitsVnot one eachVas well as four
ion-channel circuits. Indeed, it is six times larger in area
(2560 m2 ) than HICANN’s synapse (see Table 7).
VIII. DISCUSSION
We found that Neurogrid’s shared-dendrite (SD) architecture achieved the lowest AET cost by using multilevel axon
branching to increase synaptic connectivity. Its bandwidth-efficient interarray communication mechanismV
multicast tree router (TR)Vmade this possible. We
conclude by discussing how large-scale neural models
can fully exploit Neurogrid’s simulation capacity, the
limitations its architecture has, and how its configurability
and scale may be increased.
Neurogrid’s cost-effective SD architecture and TR
topology can be fully exploited by neural models that
satisfy two requirements. First, they are organized into
layers such that neighboring neurons within the same layer
have mostly the same inputs (as in cortical feature maps).
Second, they are organized into columns such that neurons
at corresponding locations in different layers have
translation-invariant connectivity (as in cortical columns).
The first requirement allows SD to be used. Otherwise, the
receiver has to cycle nsyn times to deliver the spike to nsyn
targets, instead of just once. The second requirement
14
We could not measure the power directly because USB throughput
was limited to about a million spike/s (constrained by our current CPLD
firmware).
Benjamin et al.: Neurogrid
allows multicast routing to be used. Otherwise, the
daughterboard has to cycle ncol np times to route the
spike to arbitrary locations, instead of just ncol times.
A limitation of Neurogrid’s SD architecture is it
precludes synaptic plasticity, which HICANN and other
realizations of the fully dedicated architecture (FD)
support [72], [73]. This limitation arises because neighboring neurons share the same (spatially decayed) input.
Nonetheless, Neurogrid also supports the shared-synapse
(SS) architecture, which does allow individual connection
weights (stored in the daughterboard’s RAM [69], [74]).
SS can realize spike-timing-dependent plasticity (STDP),
which HICANN realizes, by tracking a synapse’s recent
spike history (i.e., queuing address–events) and updating
the stored weight accordingly [75]. However, SS is N times
less AET-efficient than FD (see Table 1). In practice,
however, FD’s N 2 area scaling makes it prohibitively
expensive to furnish each neuron with thousands of
synapsesVHICANN has only 224 synapses per neuronV
until emerging nanoscale devices become viable [76].
Neurogrid’s configurability and scale may be increased
by migrating from its decade-and-a-half-old process to a
state-of-the-art one, which will allow the memory embedded in each Neurocore to increase by two orders of
magnitude. This additional memory will make it possible
to replace shared dendrites with local shared axons (i.e.,
tertiary branches), which offer greater configurability
while being area-efficient. And it will make it possible to
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ABOUT THE AUTHORS
Ben Varkey Benjamin received the B.Tech.
degree in electronics and communication engineering from Mahatma Gandhi University, Kerala,
India, in 2005 and the M.S. degree in electrical
engineering from Stanford University, Stanford,
CA, USA, in 2010, where he is currently working
toward the Ph.D. degree in electrical engineering.
He worked for three years as a Design Engineer
in the VSBU group at Wipro Technologies, India,
where he earned the Prodigy award for the best
incoming employee of the year. While there, he also received two U.S.
patents for his work on standard cell characterization. He led the testing
and characterization of Neurogrid, as well as the design and implementation of the software driver used to program and communicate with
Neurogrid. His research focuses on challenges in designing neuromorphic hardware for deep submicrometer technologies.
Peiran Gao received the B.A. degree in neurobiology and physics with minor in electrical engineering and computer science from the University
of California Berkeley, Berkeley, CA, USA, in 2009
and the M.S. degree in bioengineering from
Stanford University, Stanford, CA, USA, in 2011,
where he is currently working toward the Ph.D.
degree in bioengineering.
As a Teaching Assistant at Stanford University
for the BIOE 332 Large-Scale Neural Modeling
course in Spring 2011 Quarter, he developed the dynamical system
guided mapping procedure that Neurogrid uses. His research focuses on
the theoretical development of a spike-based computational framework.
Emmett McQuinn received the B.S. degree in
computer science from Clemson University,
Clemson, SC, USA, in 2008, and the M.S. degree in
computer science from the University of California
San Diego, La Jolla, CA, USA, in 2010.
He was a Research Staff Programmer at
Stanford University, Stanford, CA, USA, where he
led the development of real-time visualization
software for Neurogrid. He then joined the
Almaden Research Center, IBM, San Jose, CA,
USA, to work on the SyNAPSE project. He is currently working for a
startup. His research interests include interactive real-time visualization,
hardware-accelerated scientific computing, and scalable neuromorphic
systems.
Swadesh Choudhary received the B.Tech. and
M.Tech. degrees in electrical engineering from the
Indian Institute of Technology, Bombay, India, in
2010 and the M.S. degree in electrical engineering from Stanford University, Stanford, CA, USA,
in 2012.
He also worked as a Research Assistant at
Stanford University to develop a daughterboard
for Neurogrid. He is currently a Design Engineer at
Intel Corporation, Santa Clara, CA, USA, working in
the server development group.
Anand R. Chandrasekaran received the B.Tech.
degree in electrical engineering from the Indian
Institute of Technology, Madras, India, in 2001 and
the Ph.D. degree in neuroscience from Baylor
College of Medicine, Houston, TX, USA, in 2007.
He was a Postdoctoral Scholar in bioengineering at Stanford University, Stanford, CA, USA,
working on the Neurogrid project. He is currently
the CEO of Mad Street Den, an artificial intelligence company in Bangalore, India.
Jean-Marie Bussat (Member, IEEE) was born in
Annecy, France. He received the M.Sc. degree in
electrical engineering from ESIGLEC, Rouen,
France, in 1995 and the Ph.D. degree in electrical
engineering from the University of Paris XI, Orsay,
France, in 1998.
He joined the technical staff of the Department
of Physics, Princeton University, Princeton, NJ,
USA, in 1998, to work on the readout of the
electromagnetic calorimeter of the Compact Muon
Solenoid (CMS) experiment at CERN, Geneva, Switzerland. He joined the
Engineering Division of the Lawrence Berkeley National Laboratory,
Berkeley, CA, USA, in 2001 to design instrumentation for physics and
material science experiments. In 2007, he joined the Brains in Silicon
Laboratory, Stanford University, Stanford, CA, USA, to work on
Neurogrid.
Rodrigo Alvarez-Icaza received the B.S. degree
in mechanical and electrical engineering from the
Universidad Iberoamericana, Mexico City, Mexico,
in 1999, the M.S. degree in bioengineering from
the University of Pennsylvania, Philadelphia, PA,
USA, in 2005, and the Ph.D. degree in bioengineering from Stanford University, Stanford, CA,
USA, in 2010.
He is currently a research staff member at the
Almaden Research Center, IBM, San Jose, CA, USA,
where his research focuses on brain-inspired computers.
John V. Arthur received the B.S.E. degree (summa
cum laude) in electrical engineering from Arizona
State University, Tempe, AZ, USA, in 2000 and the
Ph.D. degree in bioengineering from the University of Pennsylvania, Philadelphia, PA, USA, in 2006.
He was a Postdoctoral Scholar in bioengineering at Stanford University, Stanford, CA, USA, as a
lead on the Neurogrid project. He is currently a
research staff member at the Almaden Research
Center, IBM, San Jose, CA, USA, working on the
SyNAPSE project. His research interests include dynamical systems,
neuromorphic and neurosynaptic architecture, and hardware-aware
algorithm design.
Vol. 102, No. 5, May 2014 | Proceedings of the IEEE
715
Benjamin et al.: Neurogrid
Paul A. Merolla received the B.S. degree in
electrical engineering (with high distinction) from
the University of Virginia, Charlottesville, VA, USA,
in 2000 and the Ph.D. degree in bioengineering
from the University of Pennsylvania, Philadelphia,
PA, USA, in 2006.
His research is to build more intelligent
computers, drawing inspiration from neuroscience, neural networks, and machine learning. He
was a Postdoctoral Scholar in the Brains in Silicon
Laboratory, Stanford University, Stanford, CA, USA (2006–2009), working as a lead chip designer on Neurogrid, an affordable supercomputer
for neuroscientists. Starting in 2010, he has been a research staff
member at the Almaden Research Center, IBM, San Jose, CA, USA, where
he was a lead chip designer for the first fully digital neurosynaptic core as
part of the DARPA-funded SyNAPSE project. His research includes lowpower neuromorphic systems, asynchronous circuit design, large-scale
modeling of cortical networks, statistical mechanics, machine learning,
and probabilistic computing.
716
Proceedings of the IEEE | Vol. 102, No. 5, May 2014
Kwabena Boahen (Senior Member, IEEE) received the B.S. and M.S.E. degrees in electrical
and computer engineering from The Johns
Hopkins University, Baltimore, MD, USA, both in
1989 and the Ph.D. degree in computation and
neural systems from the California Institute of
Technology, Pasadena, CA, USA, in 1997.
He was on the bioengineering faculty of the
University of Pennsylvania, Philadelphia, PA, USA,
from 1997 to 2005, where he held the first
Skirkanich Term Junior Chair. He is currently an Associate Professor in
the Bioengineering Department, Stanford University, Stanford, CA, USA.
He directs the Brains in Silicon Laboratory, Stanford University, which
develops silicon-integrated circuits that emulate the way neurons
compute, linking the seemingly disparate fields of electronics and
computer science with neurobiology and medicine. His contributions to
the field of neuromorphic engineering include a silicon retina that could
be used to give the blind sight, a self-organizing chip that emulates the
way the developing brain wires itself up, and a specialized hardware
platform (Neurogrid) that simulates a million cortical neurons in real time
rivaling a supercomputer while consuming only a few watts.
Dr. Boahen received several distinguished honors, including a
Fellowship from the Packard Foundation (1999), a CAREER award from
the National Science Foundation (2001), a Young Investigator Award
from the U.S. Office of Naval Research (2002), and the National Institutes
of Health Director’s Pioneer Award (2006) and Transformative Research
Award (2011). His scholarship is widely recognized, with over 80
publications to his name, including a cover story in the May 2005 issue
of Scientific American.
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