- A Summation Neuron -

This is an introduction to a circuit that I call a Summation Neuron or Ns. I have been experimenting with it off and on for quite a while now. My hope was to be able to construct a circuit that would behave and respond as much like a biological neuron as my abilities and understanding would make possible.

This is a multi-input spiking neuron. It produces a series of pulses, the number and frequency of which is related to the quantity and type of signals arriving at its inputs, and at which inputs they are received. It is modeled after biological neurons, and based on concepts and designs that have been developed by various researchers in the field of neuromorphics.

(See: "Credit to who credit is due" at the end of this article)

Of course professional researchers have access to materials, equipment and techniques that you and I do not, so it has been adapted to fall in line with the type of parts, tools and skills available to the average hobby robotics enthusiast.

As a whole, this circuit may be in violation of the K.I.S.S. (Keep It Simple Stupid) principle of most BEAM circuits, but I hope that its potential usefulness will, out weigh its involved structure. If I'm wrong, oh well, I'm learning a great deal from experimenting with it, and having a good time.

I have chosen to break the circuit down into several segments, so as to make it simpler to understand. This should also make it easier for you to make use of any part you like and set aside the rest if you will.

I have also decided; as best my knowledge permits, to give the various segments of the circuit the names of their biological counter parts. I have done the same with the terms used to describe the different functions, responses and behaviors of the various parts and the Ns as a whole.

The main divisions of the circuit in order of discussion are:

1) Synapse   2)Dendrite   3)Soma    4) Axon Hillock    5)Axon

But before we look at the circuitry, there is one characteristic of the Ns that is not normally found in BEAM neurons. I'm referring to the incorporation of what in a biological neuron is called the Resting Potential.

- I'm Just Resting -

The Resting Potential (Rp) is the normal voltage potential within a neuron when there has been no perturbation; that is to say, no signals have been received at any of its inputs.

In the summation neuron, this is simulated by a charge held by the capacitors in the circuit. High value resistors are connected between each capacitor and the source of the resting potential.

This provides a pathway for bleeding off any positive or negative (Relative to the Rp)  potentials that have been produced by signals received at the neuron's various inputs, and held by the capacitors. Thus the tendency will always be for the neuron to return to a resting state. In this way, older signals will carry a lower weight, that is have less influence, as time passes than more recent ones.

The Rp should be set somewhere between the upper and lower thresholds of the schmitt inverter used to build the Axon Hillock. From what I've learned, to make the circuit behave as much like its biological counter part as possible, the resting potential should normally be set just under 40% of the way between the positive going and negative going thresholds. Using a 74HTC14 with a 5.5 volt power source would mean setting the resting potential at about 1.22 volts.

The level of the resting potential can be set by something as simple as a pair of resistors arranged as a voltage divider, or as involved as an active regulator circuit that can adjust the resting potential according some set of criteria.

Actually, the ability to adjust the Rp is quite useful in solving a particular problem that arises from using logic inverters to build circuits that model a neuron's behaviors and responses .

Most of the attempts to model biological neurons that I have found use operational amplifiers and/or voltage comparators and have an adjustable trigger threshold. However this is not readily available when using inverters like those commonly used to build most BEAM circuits. Fortunately, the Rp provides us with an option that produces a similar type of effect.

Adjusting the resting potential of the circuit can make the axon hillock circuit more or less likely to fire. This effect is very much like having an adjustable trigger threshold.

Raising the resting potential so that it is closer to the trigger level (the inverter's positive going threshold) is equivalent to lowering the neuron's trigger threshold. Since there is less of a difference between the resting level and the trigger level, less of an increase in signal(s) is required for the neuron to fire. It will therefore fire sooner and at a higher frequency.

This can be useful in making the robot's sensory neurons more sensitive, that is, more likely to detect a small increase in sensory input in an unknown environment. Motor neurons would likewise respond faster. You might think of this as a state of heightened alertness, which could be useful when danger is detected, readying the robot to deal with conditions requiring a fight or flight response.

On the other hand, if the resting potential is adjusted downward, away from the trigger threshold, the neuron becomes less sensitive. That is, it would take a stronger signal, or a greater weight of pulses (a higher frequency, or a longer series of pulses), or the signal arriving at an input that is closer to the output of the dendrite (D-out) to make the neuron to fire. This could be thought of putting the robot in a more passive or restful state.

- Three Of A Kind -

Now I can begin to describe the individual segments of the summation neurons.

I'll start with the input gating part of the circuitry. In biological neurons this is called a synapse or synaptic connection. Presented here are three circuit variations that determine the kind of influence an incoming signal will have on the output of the neuron. They are so simple it almost seems improper to refer to them as a separate part of the circuit, but since there are three types, I think that it will be clearer if I do so.

The three types are:

Excitatory = (EXC) = Allows a positive going pulse or potential to enter a Dendrite input, thus making the neuron more likely to fire.

Click on Thumb Nail Image to View Full size image.

Inhibitory = (Inh) = Allows a negative going pulse or potential to enter a Dendrite input, thus making the neuron less likely to fire.

Resting Potential = (Rp) = Ties a Dendrite input to the resting potential source through a CMOS Bilateral Switch (1/4 of a CD1406). It is activated by an excitatory signal at the Ri input. I have come across this concept in several of the papers on neuromorphics and so have included it here as one of the available options.

Figure 1

Figure 2

Figure 3

The polarity of the first two synapse' (whether it is inhibitory or excitatory) is established by the polarity of the diodes, shown in black in the schematics. Any common type diode should work for this. (See Note at the end of artical)

The resistors provide current limiting and should be at least 1K. The value used can have an effect on how much influence an incoming signal will have on the output of the neuron, and can be changed to fine tune the effect as desired. Increasing the value of the resistor will reduce the amount of influence an  incoming signal will have on the neuron.

The incoming signal can come from something as simple as a pot or photo diode, but will usually be made up of the pulse(s) coming from another neuron, or some other circuit. The amount of influence, the potential that results from an incoming signal will have, is referred to as the weight it carries and is determined by several factors. These include the level of the signal itself, its point of input, the frequency and number of pulses if applicable and how long it has been since it arrived.

The Blue diodes (again, any common type) in the diagrams provide a means of implementing an input disable. In biological neurons this is called Pre-synaptic Inhibition (PSI). I used the abbreviation Si for Synapse inhibit. Note that the polarity of the Si diodes (gray) are opposite that of the main diodes (Black). This is one of the reasons for including the current limiting resistor.

To disable an inhibitory synapse, the Si input must be pulled high. On the other hand, to disable an Excitatory or Resting synapse, its Si input must be pulled low.

Another reason for the current limiting resistor is so that any individual input of the dendrite can receive signals from several of the various synapse' at the same time. The resulting potential on the capacitor will be an average of all of them combined together.

For instance, if a dendrite input receives signals from an Exc-synapse and an Inh-synapse at the same time, and the Inh-synapse has a resistor that is twice the value of the resistor in the Exc-synapse, the resulting effect on that input will be 2/3 of the effect of the Exc-synapse alone.

On the other hand, if two Exc-synapse' , having equal size resistors the resulting input will have twice the effect of having only one Exc-synapse, innervating, or applying signal to, the input.

So the value of the resistor in the synapse can be changed to alter how the neuron is going to respond to the signal the synapse is delivering to it.

- Need Input -

Click on Thumb Nail Image to View Full size image.

Next, let's look at the dendrite portion of the neuron. This is basically a resistor/capacitor network, that has several weighted inputs for collecting the signals from other neurons, sensors or control circuits. The resulting potentials are then passed to the next part of the circuit, which is the soma.

Figure 4

The dendrite circuit has four primary inputs labeled In-1 through In-4. It also has a fifth input, that is marked D-In. D-in allows for the creation of longer dendrites by adding more input stages to the circuit.

The basic dendrite circuit has four 1Meg resistors (R1 - R4) and four 1uf capacitors (C1 - C4) that act as individual integrators for each input. The capacitors collect and store the incoming signals. The 10 Meg resistors (R13 - R16) provide the pathway for the resting potential for each of the capacitors. The other 1 Meg resistors (R9 - R12) along with the 2.2 Meg resistors (R5 - R8) are arranged in a manner similar to an R2R network.

An R2R resistor network is sometimes used as a simple digital to analog converter. The inputs would be connected to a latch, which would be used to hold a binary number. A logic 1 (high) on an input line would result in an increased voltage at the output. A logic 0 (low) on an input line would not increase the output potential.

Input #1 has approximately twice the effect on the output potential as input #2, and input #2 has twice the effect of input #3 etc. The resulting output would be a voltage proportional to the binary number on the input latch.

The papers on silicon dendrites that I have read do not give a lot of information about component values in the circuits, used by the professional researchers. They would probably be impractical for hobbyist anyway. But it seemed to me that the dendrite circuits looked very similar to the R2R arrangement, so I figured I would try these relationships in value in my design as a starting point. Fortunately it seems to work pretty well.

The result is that the initial effect of an incoming pulse, on In-1 has a greater effect on the potential at the output (D-out), than a comparable pulse on In-4. This is similar to the way the dendrites of a biological neuron function. It is not a perfect copy by any means, but I think it will be close enough to use as a starting point.

Of course this R2R network is not receiving input from a latch, but from the circuit's capacitors instead. And the charge on the capacitor will vary with time and depend on the frequency and number of pulses being received.

An excitatory (high going) pulse at an input will increase the potential held by that input's capacitor. The result will be a greater potential at D-out. The higher the frequency, and/or the greater the number of pulses received at an input, the larger the increase in the potential on the capacitor.

If the incoming signal is made up of inhibitory (low going) pulses, the potential that appears on the output will be decreased accordingly.

If no signals are coming into the dendrite, the potential on the capacitors will begin to move toward the resting potential, and thus so will the potential that appears on D-out.

If the signal on the input is from a Rp-synapse any charge on that inputs capacitor will move more quickly, toward the neuron's resting potential. If the stored potential is higher than the resting potential, it will be pulled down toward the Rp. If it is lower, it will be pulled up toward the Rp. How much closer to the Rp the potential on the capacitor will move, depends on how many times and at what frequency the Rp-synapse is being activated.

Also, if an inhibitory and an excitatory synapse were be connected in parallel. It would then be possible to select whether inhibitory potential; excitatory potential, or no potential at all, would be allowed to pass from a sensor attached to one of the dendrite's inputs.

Need More Input

As stated earlier, the "D-in" input allows longer dendrites to be constructed. You can add extra input stages, or even another complete dendrite circuit.

It is also possible to connect the output of one complete dendrite to any of the primary inputs (1-in through 4-in) of another. Thus creating a root like structure that may be useful for processing complex interactions between different sensory input.

Normally a synapse is not necessary to do this, but one may be used if it is desired for a special purpose. For example, if it is desirable to be able to disable the input from an attached dendrite, a Rp-Synapse could be used. The additional dendrite would be connected to the Rp input of the synapse.

Theoretically, quite large and complex dendrite root structures could be created using these methods. But remember that the farther an input's potential has to travel to reach the soma, the weaker its influence will be.

If the dendrite is too large, and therefore an input is to far away from the soma, any signals received at that input could become almost inconsequential. However, my experiments seem to imply that the dendrites would have to be very large for this to be a problem.

-I Need Soma That -

Click on Thumb Nail Image to View Full size image.

Next in line is the Soma. This is the final point of summation in the neuron. It uses a 1uf capacitors, two 1 Meg (input) resistors, a 2.2 Meg (output) resistor and a 10 Meg (rest) resistor.

The potential from the dendrite is tied to the soma's primary input (D-in). It passes through resistor R17 (1Meg) to the soma's capacitor (C5). The capacitor is also tied to the resting potential through resistor R18 (10 Meg). The potential that is accumulated on C5 then passes through resistor R20 (2.2 Meg) and out to the axon.

Figure 5

There is also a synaptic input to the soma. Why you ask? Because; in biological neurons, some synaptic connections are made directly to the soma. Signals that enter the soma, rather than through a dendrite are usually inhibitory, but excitatory input is not unheard of.

Signals that enter here also pass through a 1meg resistor (R19). As with the dendrite input, the synapse' resistor can be used to influence how much effect a signal entering here will have.

Since a signal that enters the synaptic input of the soma is routed more directly to its capacitor, it will carry more weight than the signals that are collected by the dendrite. This makes the Soma's synaptic input perfect input for use when making cross connections with another neuron to prevent simultaneous triggering of both. A very useful thing when the outputs of the two neurons are being used to control a leg motor or something similar.

- I Just Nu It -

Up to this point the summation neuron has been pretty much a passive device. The Axon-Hillock,is where things start getting active. This is the point in the circuit where the output pulses of the neuron are actually produced.

Click on Thumb Nail Image to View Full size image.

Biological neurons have one output that produces a series of one type of pulse. These pulses travel along the axon and branch out to other neurons from there.

The type of synapse used at the point that a connection is made to these other neurons will determine whether the effect a pulse will have at any particular connection will be excitatory or inhibitory.

Figure 6

Now if the Axon-Hillock circuit had only one type of output, it would require the use of extra inverters in some of the synaptic circuits. Thankfully we don't have to worry about that. In fact there are two equal but opposite outputs that are available with this design.

Figure 6 shows a stand-alone Axon-Hillock, circuit that has both inhibitory and excitatory outputs. This is a side benefit of the design and it eliminates the need for using inverting in the synaptic circuit portions.

The inhibitory output produces low going pulses at the same time that the excitatory output produces high going pulses. These outputs, are the ones that will be used in the majority of BEAM applications.

There is also an output shown in Blue and marked Spike. It is tied to the point where all the key components are connected together. It is here that the neuron's Action Potential can be directly viewed with an oscilloscope.

Click on Thumb Nail Image to View Full size image.

Action Potential is the term used when referring to the pulses produced by the Axon-Hillock and Axons of biological neurons. Since I wanted this circuit to behave as much like its biological counterpart as possible, I tried to make the pulse created at the spike point of this circuit look as much like the action potential of a biological neuron as I could.

Figure 7 is an illustration of the action potential of a biological neuron.

Figure 8 shows the waveform produced by the summation Neuron.

Figure 7

What do you think ?

Click on Thumb Nail Image to View Full size image.

You'll note that the width of the action potential created by the Ns is about 100 times the width of the action potential of the biological neuron. This seemed to be a good compromise between nature and BEAM. If you would prefer to have the action potential width be closer to either, I would recommend changing the capacitors' values rather than the resistors' values. Capacitor C7 should be about half the value of C6.

Not only are there fewer components to change my experimentation show that the over all shape of the wave form is less effected this way.

Figure 8

Back to the Action

The potential coming in from the soma passes through D1 bypassing resistor R22 (2.2meg), and begins to charge capacitor C6 (.022uf).

When the charge on C6 reaches the positive going threshold of inverter I1, the output of I1 will go low. The low on the output of I1 becomes the Inhibitory output of the circuit.

The output of inverter I1 is also fed to the input of inverter I2 causing I2's output of to go high. This high output on I2 becomes the Excitatory output of the circuit.

The high on the output of I2 by-passes R23 through D3 to reach C7. C7 (.01uf) then allows a high pulse to pass through to C6 where it drives the Action Potential sharply higher.

While this is happening, the low on the output of inverter I1 also passes through feedback diode D2 and resistor R22 (2meg) to the spike junction where it begins to discharge C6. During the discharge period, the potential from the soma is prevented from reaching C6 due to the low at the point where D1, D2 and R22 are tied together. This is similar to what in biological neurons is called the Absolute Refractory Period (ARP). When the charge on C6 drops below the low going threshold of inverter I1, the output of I1 returns to a high state.

This causes the output of I2 to go low. This time the output of I2 passes through R23 before reaching C7. In the schematic R23 is shown as being 10 meg, but I'm still  experimenting with values between 2meg and 10meg. This results in a reduced low pulse passing through C7 to C6 driving the charge on C6 lower than the resting potential. While the charge on the capacitor remains below the Resting Potential it will take a higher level of input from the Soma to produce a pulse as quickly as it would if the charge on C6 was at the Rp. This is quite similar to the Relative Refractory Period (RFP) in biological neurons.

If  the Soma is not feeding excitatory or inhibitory potential to the Axon-Hillock, resistor R21 (10meg) will allow the charge on C6 to return to the resting potential.

D2 and the values of R22 and C6, with some interaction with C7, fix the width of the pulses produced by the Axon Hillock. With the values listed, and using a 74HCT14 as the inverter chip, the width of the pulses is just under 100 milliseconds wide.

D1, the value of C6, and the potential coming from the soma determine the number and frequency of the pulses that will be produced.

In experimenting with both the 74HC14 and the 74HCT14 I have found that the thresholds of the 74HCT14 let the circuit respond to smaller changes in the incoming signals and therefore produces better over all results.

The use of a variable frequency with fixed width pulses, to encode information, is called Pulse Frequency Coding (PFC). It is a form of

Pulse Width Modulation and so can also be used to control the speed of motors, the brightness of LEDs or produce sound.

- And Finally... ? -

Last but not least, we come to the axon. It may not be needed for many applications, but it makes an excellent tapped delay line for use in the sequencing of actions and behaviors. In biological neurons it carries signals from the axon of one neuron to the dendrite or soma of other neurons, or to muscles, and/or glands.

It does not act like piece of wire though. In fact what happens is more like the fuse on a firecracker. An ignition at the beginning of the fuse causes the chemicals at that point to begin to burn. They in turn ignite the chemicals that are adjacent to them and then die out. The process continues on down the length of the fuse. When this process reaches the firecracker's body, it ignites the chemicals therein and, BOOM!

A similar thing happens to the pulse created at the Axon-Hillock. As it rises, and falls, it causes a duplicate pulse to begin to rise and fall next to it. This process is repeated again and again as the pulse propagates down the length of the axon until it reaches a neuron, gland or muscle and then ... Well, you get the picture.

In the axon of the Summation Neuron, individual pulse circuits are interconnected end to end in a chain. This chain can be as long or short as need dictates. Also, other chains can be tapped off anywhere along the main chain if useful.

One possible use for this might be to produce the appropriate sequencing needed for a multi-legged robot. Especially in robots which have two or more motors in each leg.

Originally I planned on using a chain of nervous neurons (Nv's). However, some recent posts to the BEAM group convinced me that there was another way to do this. Figure 9 shows a modification of the Axon Hillock circuit.

On The Chain Gang

Click on Thumb Nail Image to View Full size image.

D1 allows only positive potential to enter. Unlike in the stand-alone Axon Hillock, the potential coming in from the soma does not have to bypass anything. Instead it is connected directly to, and begins to charge capacitor C6 (.022uf).

Figure 9

As with the Axon Hillock, when the charge on C6 reaches the positive going threshold of inverter I1, the output of I1 will go low. This low on the output of I1 becomes the Inhibitory output of the circuit.

The low on the output of I1 also pulls I2's input low which causes I2's output to go high. This of course becomes the Excitatory output. The excitatory output also passes through resistor R24 on to the next stage. R24 determines the pulse width. The schematic shows it as being 10 meg but I am experimenting with values down to 5meg.

D1 of the next stage allows only the high state from the output to enter. This begins charging capacitor C6, and the  process starts all over in that stage.

When the charge on C6 of this next stage reaches the positive going threshold of its I1 inverter, the resulting low on inhibitory output of that stage is also passes back to the previous axon stage as shown in Blue. It enters through the previous stage's reset input and then passes through feedback diode D4. This immediately discharges capacitor C6 and resets that stage. The process is repeated stage by stage to the end of the chain.

The rest of the circuit functions just as it does in the axon hillock. In fact the last stage is reset in the same manner as the axon hillock. The components shown in Red (D1, D2 and R22) perform the same function that they do in the stand-alone Axon Hillock.

Branching out

The axon of a biological neuron can branch out and travel to several different neurons. This can be done with the axon portion of the Ns as well. There is a small problem that has to be dealt with though.

The problem is that with two or more chains branching off of the output of a particular stage, you end up with two or more reset signals to deal with. This is a problem due to the issue of component tolerances.

What can happen is that one of the branches may reset the prior stage before another has been triggered, and the branch that has not been triggered will not produce pulses. The answer to this problem turns out to be quite simple.

Click on Thumb Nail Image to View Full size image.

As shown in figure 10, the inhibitory output of the first stage of the branches is not used to reset the previous stage. Instead, a special reset arrangement is used (Shown in red). The low states, present on the excitatory output when the axons have not been triggered, are passed through the D4 diodes to the input of inverter I3. Resistor R25 (1meg) is also connected between the input of I3 and the positive side of the power supply.

Figure 10

When the excitatory output of the first stage of each of the connected branches goes high, the input of I3 will no longer be held low. Resistor R25 can then pull the input high. This causes the output of I3 to go low. This low passes through D5 and discharges C6 of the stage being branched from, thus resetting that stage.

Well that about covers the main circuit design and concepts that I'm working with. It is an interesting project but since it may be more involved than some people may want to deal with, I thought I'd include a little bit simpler version of the circuit.

-Give Us A K.I.S.S.-

Keep It Simple Stupid

Figure 11 shows the schematic of the Sim-Sum Neuron (Ssn). The components shown in RED will make the circuit behave a little bit more like a biological neuron, but can be left out to make  the circuit even simpler.

Click on Thumb Nail Image to View Full size image.

The Ssn is a variation of Wilf's Nutron - majority logic neuron. The main difference is the use of feed back from the output of the inverter. It causes the Ssn to produce a series of pulses as with the Summation Neuron rather than a steady low output when triggered as with the Nutron.

Figure 11 Figure 12

The dendrite and soma of the Sim-Sum are combined into one and use only one capacitor (C1) for the summing (or integration) of the signals which arrive at the inputs. The value of the capacitor shown is 1uf, but could be larger or smaller to serve your purpose.

The input labeled In -5 represents the fact that you can add, as many inputs to the Ssn as you need. The schematic also shows the inputs marked In-1 through In-5 as having 1Meg resistors (R1-R5).

Identical signals on these inputs will have an equal effect on the potential that C1 holds. If you want an input to have a greater or lesser effect than another input, the resistor values can be altered. A higher resistance would cause signals entering at that input to have a smaller effect and a smaller resistance would result in a signal having a larger effect.

The Resting Potential can also be incorporated by tying R6 (10 Meg) between capacitor C1. This allows the capacitor to return to the resting potential if there are no pulses being received at any of the inputs.

Any charge that does collect on C1 passes through resistor R7 (2.2 meg) and out to the Axon-Hillock segment of the circuit.

The Axon-Hillock circuitry functions more or less just as it does in the Summation Neuron. Depending on whether or not you use or leave out the components shown in Red of course.

As shown in figure 12, Nv's, shown in Blue, can be used for the axon. Branching can also be done and actually is much simpler with Nv's then it is with the axon portion of the Summation Neuron. Just attach the input of the first Nv of the branch, to the output of the Nv at the point the where a branch is desired. The inverters shown in RED, can be used to provide excitatory output if needed.

The Sim-Sum Neuron works okay, but the Summation Neuron will behave more, though not exactly by any means, like a biological neuron. I'm still experimenting with it, and I haven't decided if the extra complexity is worth it yet. But personally, I tend to think it will be.

In Conclusion

Let me say that in no way do I consider the summation neuron a finished work. In fact I will be continuing to experiment with different values for the various capacitors and resistors, and circuit arrangements, as well as possible different types of inhibitory and excitatory input methods.

I also plan to experiment with the effect of feedback to the dendrites from the circuit's own excitatory and inhibitory outputs as well as from various stages of the axon.

I hope to try using variations of the axon hillock with an assortment of sensors in an attempt to create receptor neurons with behavior similar to their biological counterparts.

I Intend to write some follow up articles sharing the data I collect while testing the summation neuron along with a few more images of the action potential produced by it, as observed with an oscilloscope.

Amongst other thing I plan for these articles to  compare my collected data with the characteristics of biological neurons and give examples of useful modifications and applications for the Summation Neuron as well as its parts.

Wish me luck.

So long,

Droidmakr

Nov - 4 - 2000

NOTE: In a recent post to the BEAM eGroup, Wilf suggested that some diodes (e.g., 1N4148) may not work well with the HCT family. If the circuit does not function properly for you, try changing the diodes with a different type.

Credit to who credit is due:

The design and configuration of the circuit are an adaptation of the work of several men.

The layout of the Axon Hillock and axon are based on the work of Carver Mead,

The layout of the dendrite is based on the work of John G Elias, W. Westerman, D.P. Northmore

The techniques used to adapt the work mentioned above are based on the work of Mark Tilden.

I have borrowed heavily from the work of these people, and I am grateful to them for sharing so much of their work on the Internet.

The book "The Brains Of Men And Machines" by Ernest W. Kent, published by McGraw Hill in 1981 was the original inspiration for building a neuron circuit was.

The book "Brains, Behavior And  Robotics"; By James S. Albus, published by McGraw Hill in 1981

was also influential in my desire to build a robot that is controlled by a circuit that functions in a manner similar to a biological nervous system.

Both of these books are still listed on Amazon .com if you'er interested.