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Analog VLSI Circuits for Short-Term Dynamic Synapses
Shih-Chii Liu Institute of Neuroinformatics University of Zurich and ETH Zurich Winterthurerstrasse 190 CH-8057 Zurich, Switzerland

shih@ini.phys.ethz.ch

Abstract Short-term dynamical synapses increase the computational power of neuronal networks. These synapses act as additional lters to the inputs of a neuron before the subsequent integration of these signals at its cell body. In this work, we describe a model of depressing and facilitating synapses derived from a hardware circuit implementation. This model is equivalent to theoretical models of short-term synaptic dynamics in network simulations. These circuits have been added to a network of leaky integrate-and-re neurons. A cortical model of direction-selectivity that uses short-term dynamic synapses has been implemented with this network.

1 Introduction

Cortical neurons show a wide variety of neuronal and synaptic responses to their input signals. Networks with simplied models of spiking neurons and synapses and consisting of one or two time constants already exhibit a large number of possible operating regimes [van Vreeswijk and Sompolinsky, 1998,Brunel, 2000]. Simulations of these spiking networks can take a long time on a serial computer. In most network simulations, synapses are assumed to be static. Recent physiological data, however, show that synapses frequently show activity-dependent 1
plasticity which vary on a time scale of milliseconds to seconds. In particular, short-term dynamical synapses [Tarczy-Hornoch et al., 1998, Tarczy-Hornoch et al., 1999, Varela et al., 1997, Markram et al., 1998, Reyes et al., 1998] with time constants of hundreds of milliseconds are seen in many parts of the visual cortex. When these synapses are stimulated with a train of input spikes, the amplitude of the membrane potential of the neuron or the excitatory postsynaptic potential (EPSP) decreases (depressing synapse) or increases (facilitating synapse) with each subsequent spike. The recovery time of the maximum synaptic amplitude is in the order of hundreds of milliseconds. These synapses encode the history of their inputs and can be treated as time-invariant lters with fading memory [Maass and Sontag, 2000]. These activity-dependent synapses when added to the network allow for different forms of dynamical networks that can process time-varying patterns [Tsodyks et al., 1998, Maass and Zador, 1999]. Examples of how these synapses could contribute to visual cortical responses include direction selectivity [Chance et al., 1998] and automatic gain control [Abbott et al., 1997]. The simulation time of spiking networks with different types of activity-dependent synapses consisting of different time constants will increase signicantly. This simulation time can be shortened by using a hardware implementation of a network with spiking neurons and these activity-dependent synapses. Here, we describe a circuit model of short-term synaptic dynamics based on the silicon implementation of synaptic depression and facilitation by [Rasche and Hahnloser, 2001]. The dynamics of this circuit model is qualitatively comparable to the dynamics of two theoretical models [Boegerhausen et al., 2003]; the phenomenological model from [Markram et al., 1998, Tsodyks and Markram, 1997, Tsodyks et al., 1998] and the model from [Abbott et al., 1997, Varela et al., 1997]. Measurements from these circuits on a fabricated chip show how these synapses lter the inputs to a leaky integrate-and-re neuron under transient and steady-state conditions. The dynamics of short-term plastic synapses are dependent on the frequency of the presynaptic input. In the case of a neuron which is stimulated through a depressing synapse by a regular input spike train, the ring rate of the neuron decreases over time due to the decrease in synaptic input with each presynaptic spike. Interestingly, a class of neurons in the cortex also adapt their ring rate over time in response to a regular spike input through a normal synapse. This output adaptation mechanism is non-input specic whereas the rst mechanism involves the ltering of specic inputs. The inclusion of these short-term synapses into networks of neurons allow pro2
cessing of time-varying inputs. However, the simulation time of such networks on a computer increases substantially as more different types of time constants are added to the circuits. The previous constructions of neuron circuits ranging from Hodgkin-Huxley models of neurons [Mahowald and Douglas, 1991, Patel and DeWeerth, 1997] to integrate-and-re neurons [Mead, 1989,Sarpeshkar et al., 1992, Lazzaro and Wawrzynek, 1994, Boahen, 1997a, van Schaik, 2001], together with long-time constant learning synapses [Hasler et al., 1995, H iger and Maa howald, 1999] and short-term dynamic synapses [Rasche and Hahnloser, 2001] can be used to develop realistic, real-time, low-power, spike-based networks.

2 Synapses

Synaptic circuits have been implemented using very few transistors [Boahen, 1997b, Rasche and Hahnloser, 2001]. However, their dynamics are usually different from the exponential dynamics of synaptic models used in simulations. To implement the exponential dynamics, we would have to use a linear resistor to obtain the exponential dynamics. A transistor can act as a linear resistor as long as the terminal voltages satisfy certain criteria. Additional circuitry would be needed to satisfy these criteria thus increasing the nal size of the circuit. One alternative is to replace the linear resistor dynamics with diode dynamics which is easily obtained with one diode-connected transistor. We will discuss the difference between the diode-connected transistor dynamics and the exponential dynamics for the different types of synapses.

2.1 Normal Synapses

In simulations, the synaptic current at the time of the spike :

0 1)

is either treated as a point current source
is a xed current or as a current source with a nite decay time:

CA @4 ! 2 (B9753$ ('

where is the time constant of the decay and is measured right after a spike. The point current source can be implemented by two transistors (for example, and in Fig. 1(a)). If we need a synaptic current with a nite decay time,

H GE F GE

! &%$#"
we include the current-mirror circuit, , , and. Unlike the dynamics in Eq. 1, the synaptic current has a decay dynamics [Boahen, 1997b] rather than exponential dynamics. The decay of is described by
S i ph g fd b ` Beca2 X T Y1 T W V &W2 T 7 T U

Ir Vd M2

Figure 1: Current-mode circuits for a normal synapse (a) and a depressing synapse (b). The detailed operation of the circuits is described in the text.
2.2 Short-Term Synaptic Dynamics
Dynamical synapses can be depressing, facilitating, or a combination of both. In a depressing synapse, the synaptic strength decreases after each spike and recovers towards to its maximal value with a time constant,. In facilitating synapses, the strength increases after each spike and recovers towards to its minimum value with a time constant,. Two prevalent models that are used in network simulations and also for tting physiological data are the phenomenological model by [Markram 4

r t uPS

T r sq

where value of

, at the time of the spike
is the thermal voltage, and.

Vgain Va M1 M6

Vx M5 C Vpre Isyn M4

X T YW

y i $pw y x 4 f C f

is the

et al., 1998, Tsodyks and Markram, 1997, Tsodyks et al., 1998] and the model from [Abbott et al., 1997, Varela et al., 1997]. We only consider the dynamics of the model from Abbott and colleagues in this work. 2.2.1 Simulation Model of Short-Term Dynamic Synapses The dynamics of the depressing synapse is similar to the adaptation dynamics of the photoreceptor. Both elements code primarily changes in the input rather than the absolute level of the input. The photoreceptor amplies the contrast of the visual signal and has a low gain to background illumination. The output of the depressing synapse codes primarily changes in the presynaptic frequency. The synaptic strength adapts to a steady-state value that is approximately inversely dependent on the input frequency. Thus, the depressing synapse acts like a band-pass lter to spike rates much like the photoreceptor has a band-pass response to illumination. The facilitating synapse, on the other hand, acts like a low-pass lter to changes in spike rates. A step increase in presynaptic ring rate leads to an increase in the synaptic strength. Both types of synapses can be treated as time-invariant fading memory lters [Maass and Sontag, 2000].

M4 Isyn M5

Id M6 C2 M7
Figure 2: Synaptic facilitation circuit. The circuit on the left is the same as part of the circuit in Fig. 1(b). The voltage determines the synaptic strength and the current goes to the neuron. This circuit would be have to be inverted so that it can be combined with the neuron circuit in Fig. 3. In the theoretical model from Abbott and colleagues, the depression in the 5
synaptic strength is dened by a variable varying between 0 and 1. The synaptic strength is given by where is the maximum synaptic strength. The recovery dynamics of is described by:

! 2 3c T ) a l

where is
where ( ) is the amount by which is decreased right after the spike. In the case of a regular spike train, the average steady-state value of is
k A ih Wj f &@ ! 32 I j A ih f 7@ 4 ! I f g d d e
. The In the facilitating case, the facilitation is dened by a variable synaptic strength is where is the maximum synaptic strength. The recovery dynamics of is:

m nl l ! 2 3o l ) l

where is the time constant in which recovers exponentially back to 1. The update dynamics is now additive instead of subtractive:

q ` rp 6 l l $

where ( ) is the amount by which is increased right after the spike. The variable is updated additively because multiplicative facilitation can lead to increases of synaptic strength without bounds especially at high frequencies for the recovery dynamics in Eq. 4. 2.2.2 Circuit Model of Short-Term Dynamic Synapses As before, we replace the exponential dynamics in Eq. 1 with the diode-connected transistor dynamics. This replacement gives rise to the synaptic depressing circuit in Fig. 1(b) which was proposed in [Rasche and Hahnloser, 2001]. The new circuit gives rise to the following recovery dynamics for the depressing variable :
u t U@ I ! 32 s E l l d
is the recovery time constant of the depression and the update dynamics (2)
Vb Vthresh Id Ileak Vleak Cm Ipw M6 Vo M5 Vm Vpw Vm Vo Vo M4 Vrefr Ca

M1 M2 M3 Vt

Vo Vca
Spike adaptation circuitry
Figure 3: Schematic of the leaky integrate-and-re neuron. The parameters, sets the refractory period, sets the threshold voltage, sets the pulse sets the leak current. The circuit within the dashedwidth of the spike, dotted inset implements the spike adaptation mechanism. The parameters and set the adaptation dynamics. where is the equivalent of and is a transistor parameter which is less than 1 in subthreshold operation. The update dynamics are similar to Eq. 2:
k %6  %$ ~ | " z } | v { T ) W2 V x wy8 v i x p } | { $Bv E 8
2.2.3 Depressing Circuit The detailed analysis leading to Eqs. 6 and 7 for is described in [Boegerhausen et al., 2003]. The voltage determines the maximum synaptic strength while the synaptic strength or is exponential in the voltage,. The subcircuit consisting of transistors, , , and , control the dynamics of. The presynaptic input goes to the gate terminal of which acts like a switch. During a presynaptic spike, a quantity of charge (determined by ) is removed from the node. In between spikes, recovers towards through the diode-connected transistor,. Also during the presynaptic spike, transistor turns on and the ows into the membrane potential of the neuron. We can synaptic current convert the current source into an equivalent current with some gain and a time constant through the current-mirror circuit consisting of , , and the capacitor , and by adjusting the voltage. The synaptic strength is given by where 7
$W #E E Q PE T T U @ 4 i Yt B$| 0 | H E H GE FGE E I $W | U $( I E F S p

i wi v

0.35 0.3

Update

Slow recovery

V =0.3V

Fast recovery

Vx (V)

0.15 0.1 0.04 0.06 0.08 0.1 0.12 Time (s) 0.14 0.16

0.15 0.04

Figure 4: Response of to a regular spiking input of 20 Hz with different values of. (a) Change of over time. It is decreased when an input spike arrives and it recovers back to the quiescent value at different rates dependent on its distance from the resting value of about 0.33 V. (b) The steady-state value and dynamics of can be tuned by changing. Because it is difcult to compute a closed-form solution for Eq. 6 for any value and solve for after a spike of , we look at a simple case where 1. The actual value of changes for different operating has occurred at conditions and also depends on fabrication parameters. The recovery equation in Eq. 8 includes the current dynamics of the diode-connected transistor ( in Fig. 1(b)) in the region when is close to the maximum value. The equation for is then
I E 8 YF YF 4 ` 2 a! 8 YF YF 4 X Y ` X Y  k  F X Y ! 2 $ E V  T p e T

Note that if

=1, then the equation reduces to Eq. 1.
is not close to its maximum value of 1, we can approximate the dynamics (regardless of ) and solve for : (9)

E %Pu E DY ` X 4 ` 2 Pa` X Y ` X pDY !

X E %DY

The recovery time constant (

is set by

E 1

is dened as

Vd=0.24V 0.26V 0.28V
i h f t h i Y@ j 6 f h j 6 I pt d f h 4 i Y@ j 6 f t
0.1 0.12 Time (s) 0.14 0.16 0.18
V E Wj h d y d i y f x v f

@ 4 i Yet B1

2.2.4 Model of Facilitating Synapse The schematic for the facilitating synapse is shown in Fig. 2. The difference in this circuit from the depressing synaptic circuit is that the node goes to the gate of a pFET instead of an nFET. The synaptic strength is now and is directly proportional to the current variable, , so
f h 4 i Y@ j 6 f (t W1s f h 4 i Y@ j 6 f (t W1$7 p i I l (
where and. The update dynamics is multiplicative instead of additive as in Abbotts model:

6 8 j h l q $ l l

The circuit model for facilitation is quite dissimilar to Eqs. 4 and 5. Even though the update is multiplicative, the variable will not increase without bounds because the recovery dynamics of the diode-connected transistor which is a negativefeedback element. In Section 5, we will see that the steady-state value of is approximately linear in the presynaptic rate,. 9

k Y X

However, if

l ! E V l

is far from its resting value of 1, we obtain the simpler dynamics and solve for : (14)
` X E ePp E Y l l 4 ` 2 Pa` X Y ` X pDY !

4 ` 2 a! 8

X E eY

4 X DY

` X pY

where Using

. In steady-state, , the equation for is

2 ! t @ I 2 3F

k ' i h f t h i Y@ j 6 f h j 6 I pt d
. The recovery dynamics is given by (12).

f h 4 i Y@ j 6 f Ut W

` 2 aoq

In this regime, when.

T ) d d
follows a linear trajectory. Note that the same is true of Eq. 1

3 V (V)

Vm (V)

0.2 Time (s) 0.3 0.4

0.0.15 0.2 0.25 0.3 Time (s) 0.35 0.4
Figure 5: Transient response of a neuron (by measuring its membrane potential, ) when stimulated by a regular spiking input through a depressing synapse (a) and a facilitating synapse (b). The leak current of the neuron has been adjusted so that the neuron does not reach threshold. (a) The EPSP decreases with each incoming input spike for a depressing synapse. (b) The EPSP increases with each incoming input spike for a facilitating synapse. The initial EPSPs are not seen because the leak current of the neuron is larger than the synaptic current.

3 Neuron Circuit

The dynamics of the neuron circuit are similar to that of a leaky integrate-and-re neuron with a constant leak, Fig. 3. The circuit is described in detail in [Indiveri, 2000, Liu et al., 2001]. It is a modied version of previous designs [Mead, 1989, van Schaik, 2001] and also includes the circuitry which models ring-rate adaptation [Boahen, 1997a, Boahen, 1997b] frequently seen in pyramidal cells. The equation for the depolarization of the soma is as follows:
x y8 v i x d x $| 5! } | $Bv { ! #' p
where is the synaptic current to the soma, is the leakage current, and is the after-hyperpolarization potassium ( ) current which causes the adaptation in the ring rate of the cells. When increases above at ( is the time of spike), it increases by a step increment which is determined by the capacitive coupling and. The output becomes active at this time and turns on the discharging current path through transistors and. The time during which remains 10
I x $| S $ $ } | $Bv { G GE x wy8 v i x E p S

Vd = 0.2 V

Vf = 0.35 V

V = 0.4 V

V = 0.3 V

Vf = 0.45 V

Vd = 0.35 V

0.2 Time (s) 0.4

0 0.15

0.3 0.35 Time (s)

Figure 6: Transient response of a neuron to a regular spiking input for various values of and. (a) The amount by which each EPSP depresses for each subsequent pulse is set by. (b) The amount by which each EPSP facilitates is set by. high, , depends on the time taken for to discharge below. In this design, the pulse width is determined by the rate at which is discharged which in turn depends on the difference between the input current , the leak current , and the current. In other designs, is reset immediately below when becomes active because either the input current is blocked from charging the membrane or the current is much larger than the input current. The refractory period, is determined by which keeps high so that cannot charge up the membrane. The spike output is taken from the node. The time taken for the neuron to charge up to threshold is (16)
T U T & T U }v { | ! 5p S x wy8 I v i x i wi v z ` S z d u } | $Bv { x wy8 v i x x wy8 v i x T T
and in the case of a constant input current

` d I E

, the spike rate is (17)

k | S `

to and the capacitor in Fig. 3 impleSpike Adaptation Transistors ment the spike adaptation mechanism. The data in Fig. 8(b) show the adaptation of the output spike rate when the neuron was driven by a 100 Hz regular input 11

1.4 1.2

Vm (V)

5 Spike number 10 15

0.8 0.6 0.4

0.05 0

Spike number
Figure 7: Change in the membrane potential for two different settings of and. (a) There is initial facilitation of the EPSPs before depression. = 0.75 V and = 0.2 V. (b) Only depression is seen in the EPSPs. = 0.817 V and = 0.3 V. spike train through a non-plastic synapse. The amount of charge dumped on is determined by. The dynamics of the current mirror circuit ( , , and ) are used to set the dynamics of the current. The adapted spike rate is reduced from the initial rate by a factor [Boahen, 1997a] where is the charge that is dumped onto the capacitor during each postsynaptic spike (that is, when is high), is the amount of charge needed for to reach threshold, and.

| | S T T | |

4 Transient Response
The data in the gures in the remainder of the paper are obtained from a multineuron circuit with depressing and facilitating synapses fabricated in a 0.8 m CMOS process. To show the effect of synaptic depression, we measured over time as the input was driven by a regular spike train as shown in Fig. 4. Remember that the synaptic strength, is exponential in. When there are no spikes, is approximately equal to. During a spike, is decreased by an amount dependent on. This node recovers in-between spikes at a rate that depends on the difference in voltage between and. The recovery rate is faster when is far from. The dependence of the recovery rate on this difference is due to 12

p | p | T |

x y8 q

` 2 $ x $|

the current-mirror circuit dynamics. The parameter controls both the synaptic strength and the recovery time constant. For a xed , the dynamics and the steady-state value of can be set by changing (or ) as shown in Fig. 4(b). The subsequent effect on the neuron is seen by measuring the EPSP response when a presynaptic spike occurs. The EPSPs recorded when the neuron was stimulated by a regular spiking input through these synapses are shown in Fig. 5. The parameters of the synapse and the neuron have been tuned so that the EPSPs do not add up with each incoming spike. In Fig. 5(a), the EPSP amplitude decreases with each incoming spike, while in Fig. 5(b) the amplitude increases instead. The EPSPs in response to the rst few spikes in (b) are not observable because the leak current is larger than the synaptic current. The amplitude reaches a steady-state value after a nite number of spikes. The number of spikes needed to reach steadystate can be tuned by the parameters, and. Different and values lead to different amounts of depression and facilitation as shown in Fig. 6. The ts between the circuit model and the simulation model are described in [Boegerhausen et al., 2003].

Membrane potential (V)

0.4 0.6 Time (s)
Figure 8: Mechanisms for spike frequency adaptation. (a) Adaptation due to synaptic depression. Different adapted rates are obtained by using = 0.2 V (top curve) and = 0.1 V (bottom curve). (b) Adaptation due to different after-hyperpolarization currents ( = 4.3 V (top curve) and = 4.5 V (bottom curve)). The sharp excursions of the membrane potential represent the output spikes of the neuron.

T | | T

4.1 Depression and Facilitation
We can obtain a combination of facilitation and depression dynamics in from the depressing synaptic circuit in Fig. 1(b) by choosing certain circuit parameters. The output of the current-mirror synaptic circuit in Fig. 1(a) can produce paired-pulse facilitation [Boahen, 1997b]. The equation for is the same as Eq. 10 for in the facilitating synapse circuit: where , is , and is dened as This equation also applies to in in Fig. 1(b). The difference between both circuits is that the factor that determines the change in right after a spike is
i &x 4 T & T #q T l T W i T T &x y y v d f i wC f 8 f j h d i T q &T l f h 4 i Y@ j wC 6 f (t W1 $( T
constant in one circuit, and varies for the other circuit ( ). In the depressing synaptic circuit, is not constant and depends on the input spike activity whereas in the current-mirror synaptic circuit, is constant. So for certain parameter settings in the depressing circuit, the EPSPs show initial facilitation before depressing in response to a step input of a regular 100-Hz spike train as shown in Fig. 7.
4.2 Depression or Adaptation
Both the synaptic depression and spike adaptation mechanisms lead to adaptation in the neurons ring rate to a step increase in the input rate as shown in Fig. 8. In fact, the transient response to a step increase in the rate of a regular spiking input is almost indistinguishable using either mechanism. Although both mechanisms lead to gain control in the neuron, the individual mechanisms are sensitive to different signals. Synaptic depression gives rise to sensitivity in input rate changes whereas spike adaptation makes the neuron sensitive to changes in the neurons output rate. For example, if one of the inputs to the neuron is highly active, the spike adaptation mechanism of the neuron reduces its sensitivity to the continuous large input current regardless of the origin of the large input. On the other hand, the synaptic depressing mechanism only turns down the sensitivity of that particular active input so the neuron is still selective to all other inputs. The role of depression and facilitation in implementing gain control has been described by [Abbott et al., 1997, Senn et al., 1998, Matveev and Wang, 2000].
f h 4 i Y@ j f 6 f t l T &T T i

fin (Hz)

Figure 9: Average steady-state EPSP amplitude versus input frequency in the case of a depressing synapse. The curve shows an inverse dependence of the amplitude on the frequency.

5 Steady-State Response

and on the presynaptic freThe dependence of the steady-state values of quency can be determined easily in the case of a regular spiking input. In the case of depression, we use Eqs. 7 and 8 to compute the steady-state value of :

F i B@ YF 4 ` 2 aY F ! 2 3$` l F i B@ YF YF 4 ` 2 ! $ p B@ ` i YF 4 ` 2 Y ` 2 ! $ w
Thus the steady-state EPSP amplitude is inversely dependent on the presynaptic rate as shown in Fig. 9. The form of the curve is similar to the results obtained in the work of [Abbott et al., 1997] where the data can be tted with Eq. 3. 15

! 2 3$

For the simpler dynamics of obtain a simpler expression for
, we use Eq. 9 instead of Eq. 8 and

4 ` 2 ! $ E

In the case of facilitation, we use Eqs. 11 and 13 to compute the steady-state value, :
F i B@ YF 4 ` 2 Pa$ F q ! 2 5Y` F i B@ i B@ YF YF 4 ! 2 q 3Ya 4 ` 2 ! q P$" ` i p B@ B l V YF l 4 ` 2 q ` 2 ! Pa$raY l B wB l l
In the simpler case where

, (21)

which shows that the steady-state value of is linear in the presynaptic rate and it does not increase without bounds as in the case of the exponential dynamics model for.
6 Direction Selectivity using Short-Term Synaptic Depression
Depressing synapses have been implicated in the appearance of certain visual cortical cell responses, for example, direction-selectivity. Because these synapses act like a high-pass lter in the frequency domain, the response of the neuron shows a phase advance over its response if stimulated through a non-plastic synapse. This feature was exploited in a model that described the direction-selective responses of visual cortical neurons [Chance et al., 1998]. In this model, the neuron was driven by the outputs of a set of cells in the lateral geniculate nucleus (LGN) through depressing synapses and the outputs of a spatially shifted set of LGN cells through non-depressing synapses. We have attempted the same experiment by driving a cortical neuron on our chip with spikes recorded from an LGN cell in the cat visual cortex during stimulation with a drifting sinusoidal grating (courtesy of K. Martin) and a temporally shifted version of these spikes. An example of the direction-selective response is shown in Fig. 10 [Liu, 2001]. The direction-selective results were qualitatively similar to the data in [Chance et al., 1998]. This chip has been used for exploring other spike-based cortical models, for example, orientation selectivity [Liu et al., 2001].

2 ! q WB E q

0 0.1.5 2

Time (s)

Figure 10: Response to a drifting 1-Hz sinusoidal grating. The spikes from an LGN cell in the cat visual cortex in response to the drifting grating are depicted at the bottom of the curve. The spikes from a putative spatially shifted LGN cell were generated from these spikes by shifting them in time by 60 ms. The top curve shows the response of the silicon cortical neuron when the stimulus drifted in the preferred direction. The sharp excursions at the top of the potential are the output spikes of the neuron. The middle curve shows the response to the stimulus in the null direction. The membrane potential did not build up to threshold. Figure adapted from Fig. 8 in [Boegerhausen et al., 2003] with permission.

7 Conclusion

The addition of short-term dynamical synapses to neuronal networks increases the computational power of such networks especially in processing time-varying inputs. Because of the similarity of the dynamics of the silicon models to the theoretical models, a silicon network of leaky integrate-and-re neurons which incorporate these synapses can provide an alternative to network simulations on the computer. This type of spike-based network runs in real-time and the computational time does not scale with the size of the network. This chip is a basic module in a recongurable, rewireable, spike-based system that provides ease for prototyping computational models. The system can also be useful for possible 17
applications, for example, in interfacing with neural wetware.

Acknowledgments

I acknowledge Pascal Suter and Malte Boegerhausen for some of the chip data and the simulation results in this paper. I also acknowledge Kevan Martin, Pamela Baker, and Ora Ohana for discussions on dynamic synapses. This work was supported in part by the Swiss National Foundation Research SPP grant.

References

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