TheAnalogThing - User contributions [en]

Hybrid Computer

2021-10-14T13:46:53Z

Ulmann: /* Commands */

= Hybrid computers =

A '''hybrid computer''' is a computer consisting of an [[analog computer]] and a digital computer. The two worlds are connected with ADCs/DACs (analog to digital/digital to analog converters).

[[Category:Fundamentals]]

== A simple Arduino based hybrid controller ==

[[File:Hc_setup.jpg|center|500px]]

This section describes a simple yet quite powerful hybrid computer consisting of THE ANALOG THING and an attached Arduino Mega 2650 micro controller board. Since THE ANALOG THING features a dedicated connector for attaching an external digital computer, only a few connections between this HYBRID connector and the Arduino are required to setup the hardware:

{| class="wikitable"
|+ Caption text
|-
! HYBRID pin !! description !! destination on Arduino
|-
| 2 || analog x output || AnalogIn 0
|-
| 4 || analog y output || AnalogIn 1
|-
| 6 || analog z output || AnalogIn 2
|-
| 8 || analog u output || AnalogIn 3
|-
| 9+10 || GND || GND
|-
| 13 || enable hybrid mode || D2
|-
| 14 || MOP || D3
|-
| 16 || MIC || D4
|}

These connections were made using a piggy back board suitable for an Arduino Mega 2650 which only contains a 2x8 pin header and the connections listed in the above table:

[[File:Arduino_hc.jpg|center|500px]]

(As tempting as it may be to power THE ANALOG THING by the 5V line of the Arduino - this does not work since the current requirement of the THAT is too high at about 300 mA which causes a substantial voltage drop, leaving only about 4.4 V at THE ANALOG THING which is not quite enough for correct operation. Thus, THE ANALOG THING should always be powered by a dedicated power supply with a USB-C cable while the Arduino is powered by its own USB-connection to the host computer.)

== Software ==

The required software for this hybrid controller can be found at [https://github.com/anabrid/hardware/blob/main/the-analog-thing/arduino_2650_hybrid_controller/simple_hybrid_controller/simple_hybrid_controller.ino]. It requires two additional libraries, TimerThree and TimerFive, to compile. These two timer libraries are necessary for the accurate timing of operating modes and data logging and must be installed for your Arduino IDE before compiling the source.

== Commands ==

* arm: Arm the data logger. Data logging can be performed during a single run (see below) and will automatically start when the logger has been armed before starting the single run. After completing the single run, the logger is disarmed again.
* channels=value: Set the number of analog channels to be logged. The value can range between 1 and 4.
* disable: Disable the hybrid controller. This is the normal startup mode; THE ANALOG THING will work as a standalone analog computer when the hybrid port is disabled.
* enable: Enable the hybrid controller. In this case, THE ANALOG THING is is under control of the attached microcontroller and the mode switch is disabled.
* halt: Put THE ANALOG THING into HALT-mode.
* help: Print a short help text.
* ic: Set THE ANALOG THING to IC-mode.
* ictime=value: Set the IC-time to value milliseconds. This is necessary for any single or repetitive run. Typically the duration of IC-mode is a few milliseconds. (If there are any integrators in an analog program with their time scale factor set to SLOW, the IC-time should be high enough so that these integrators can settle to the selected initial condition.)
* interval=value: Set the sampling interval for the data logger to value milliseconds.
* op: Set THE ANALOG THING to OP-mode.
* optime=value: Set the OP-time to value milliseconds.
* rep: Start repetitive operation. THE ANALOG THING will from now on cycle between IC and OP with the respective times set by the ictime and optime commands. Repetitive operation is ended when the computer is explicitly set into IC-, OP- or HALT-mode.
* run: Start a single run with the IC- and OP-phase duration determined by the times set with the ictime and optime commands. Only during such a single run data logging is possible.
* status: Return status information.

== Typical operation ==
A typical sequence of commands to run an analog program once while collecting
data looks like this:

ictime=50
optime=3000
interval=5
channels=1
arm
enable
run

This will print the AD converted values during the single run. These can then
be copied to a file (by copy/paste) and plotted with gnuplot or the like.

The analog program shown in the picture at the beginning is a Hindmarsh-Rose model of a bursting and spiking neuron. [https://analogparadigm.com/downloads/alpaca_28.pdf] Plotting the data captured with the sequence of commands shown above with gnuplot yields a result like this:

[[File:Hindmarsh_rose_x.jpg|center|500px]]

Hybrid Computer

2021-10-14T13:42:04Z

Ulmann: /* A simple Arduino based hybrid controller */

= Hybrid computers =

A '''hybrid computer''' is a computer consisting of an [[analog computer]] and a digital computer. The two worlds are connected with ADCs/DACs (analog to digital/digital to analog converters).

[[Category:Fundamentals]]

== A simple Arduino based hybrid controller ==

[[File:Hc_setup.jpg|center|500px]]

This section describes a simple yet quite powerful hybrid computer consisting of THE ANALOG THING and an attached Arduino Mega 2650 micro controller board. Since THE ANALOG THING features a dedicated connector for attaching an external digital computer, only a few connections between this HYBRID connector and the Arduino are required to setup the hardware:

{| class="wikitable"
|+ Caption text
|-
! HYBRID pin !! description !! destination on Arduino
|-
| 2 || analog x output || AnalogIn 0
|-
| 4 || analog y output || AnalogIn 1
|-
| 6 || analog z output || AnalogIn 2
|-
| 8 || analog u output || AnalogIn 3
|-
| 9+10 || GND || GND
|-
| 13 || enable hybrid mode || D2
|-
| 14 || MOP || D3
|-
| 16 || MIC || D4
|}

These connections were made using a piggy back board suitable for an Arduino Mega 2650 which only contains a 2x8 pin header and the connections listed in the above table:

[[File:Arduino_hc.jpg|center|500px]]

(As tempting as it may be to power THE ANALOG THING by the 5V line of the Arduino - this does not work since the current requirement of the THAT is too high at about 300 mA which causes a substantial voltage drop, leaving only about 4.4 V at THE ANALOG THING which is not quite enough for correct operation. Thus, THE ANALOG THING should always be powered by a dedicated power supply with a USB-C cable while the Arduino is powered by its own USB-connection to the host computer.)

== Software ==

The required software for this hybrid controller can be found at [https://github.com/anabrid/hardware/blob/main/the-analog-thing/arduino_2650_hybrid_controller/simple_hybrid_controller/simple_hybrid_controller.ino]. It requires two additional libraries, TimerThree and TimerFive, to compile. These two timer libraries are necessary for the accurate timing of operating modes and data logging and must be installed for your Arduino IDE before compiling the source.

== Commands ==

* arm: Arm the data logger. Data logging can be performed during a single run (see below) and will automatically start when the logger has been armed before starting the single run. After completing the single run, the logger is disarmed.
* channels=value: Set the number of analog channels to be logged. The value can range between 1 and 4.
* disable: Disable the hybrid controller. This is the normal startup mode; THE ANALOG THING will work as a standalone analog computer when the hybrid port is disabled.
* enable: Enable the hybrid controller. In this case, THE ANALOG THING is is under control of the attached microcontroller and the mode switch is disabled.
* halt: Put THE ANALOG THING into HALT-mode.
* help: Print a short help text.
* ic: Set THE ANALOG THING to IC-mode.
* ictime=value: Set the IC-time to value milliseconds. This is necessary for any single or repetitive run. Typically the duration of IC-mode is a few milliseconds. (If there are any integrators in an analog program with their time scale factor set to SLOW, the IC-time should be high enough so that these integrators can settle to the selected initial condition.)
* interval=value: Set the sampling interval for the data logger to value milliseconds.
* op: Set THE ANALOG THING to OP-mode.
* optime=value: Set the OP-time to value milliseconds.
* rep: Start repetitive operation. THE ANALOG THING will from now on cycle between IC and OP with the respective times set by the ictime and optime commands. Repetitive operation is ended when the computer is explicitly set into IC-, OP- or HALT-mode.
* run: Start a single run with the IC- and OP-phase duration determined by the times set with the ictime and optime commands. Only during such a single run data logging is possible.
* status: Return status information.

== Typical operation ==
A typical sequence of commands to run an analog program once while collecting
data looks like this:

ictime=50
optime=3000
interval=5
channels=1
arm
enable
run

This will print the AD converted values during the single run. These can then
be copied to a file (by copy/paste) and plotted with gnuplot or the like.

The analog program shown in the picture at the beginning is a Hindmarsh-Rose model of a bursting and spiking neuron. [https://analogparadigm.com/downloads/alpaca_28.pdf] Plotting the data captured with the sequence of commands shown above with gnuplot yields a result like this:

[[File:Hindmarsh_rose_x.jpg|center|500px]]

Hybrid Computer

2021-10-14T09:17:27Z

Ulmann:

= Hybrid computers =

A '''hybrid computer''' is a computer consisting of an [[analog computer]] and a digital computer. The two worlds are connected with ADCs/DACs (analog to digital/digital to analog converters).

[[Category:Fundamentals]]

== A simple Arduino based hybrid controller ==

[[File:Hc_setup.jpg|center|500px]]

This section describes a simple yet quite powerful hybrid computer consisting of THE ANALOG THING and an attached Arduino Mega 2650 micro controller board. Since THE ANALOG THING features a dedicated connector for attaching an external digital computer, only a few connections between this HYBRID connector and the Arduino are required to setup the hardware:

{| class="wikitable"
|+ Caption text
|-
! HYBRID pin !! description !! destination on Arduino
|-
| 2 || analog x output || AnalogIn 0
|-
| 4 || analog y output || AnalogIn 1
|-
| 6 || analog z output || AnalogIn 2
|-
| 8 || analog u output || AnalogIn 3
|-
| 9+10 || GND || GND
|-
| 13 || enable hybrid mode || D2
|-
| 14 || MOP || D3
|-
| 16 || MIC || D4
|}

These connections were made using a piggy back board suitable for an Arduino Mega 2650 which only contains a 2x8 pin header and the connections listed in the above table:

[[File:Arduino_hc.jpg|center|500px]]

== Software ==

The required software for this hybrid controller can be found at [https://github.com/anabrid/hardware/blob/main/the-analog-thing/arduino_2650_hybrid_controller/simple_hybrid_controller/simple_hybrid_controller.ino]. It requires two additional libraries, TimerThree and TimerFive, to compile. These two timer libraries are necessary for the accurate timing of operating modes and data logging and must be installed for your Arduino IDE before compiling the source.

== Commands ==

* arm: Arm the data logger. Data logging can be performed during a single run (see below) and will automatically start when the logger has been armed before starting the single run. After completing the single run, the logger is disarmed.
* channels=value: Set the number of analog channels to be logged. The value can range between 1 and 4.
* disable: Disable the hybrid controller. This is the normal startup mode; THE ANALOG THING will work as a standalone analog computer when the hybrid port is disabled.
* enable: Enable the hybrid controller. In this case, THE ANALOG THING is is under control of the attached microcontroller and the mode switch is disabled.
* halt: Put THE ANALOG THING into HALT-mode.
* help: Print a short help text.
* ic: Set THE ANALOG THING to IC-mode.
* ictime=value: Set the IC-time to value milliseconds. This is necessary for any single or repetitive run. Typically the duration of IC-mode is a few milliseconds. (If there are any integrators in an analog program with their time scale factor set to SLOW, the IC-time should be high enough so that these integrators can settle to the selected initial condition.)
* interval=value: Set the sampling interval for the data logger to value milliseconds.
* op: Set THE ANALOG THING to OP-mode.
* optime=value: Set the OP-time to value milliseconds.
* rep: Start repetitive operation. THE ANALOG THING will from now on cycle between IC and OP with the respective times set by the ictime and optime commands. Repetitive operation is ended when the computer is explicitly set into IC-, OP- or HALT-mode.
* run: Start a single run with the IC- and OP-phase duration determined by the times set with the ictime and optime commands. Only during such a single run data logging is possible.
* status: Return status information.

== Typical operation ==
A typical sequence of commands to run an analog program once while collecting
data looks like this:

ictime=50
optime=3000
interval=5
channels=1
arm
enable
run

This will print the AD converted values during the single run. These can then
be copied to a file (by copy/paste) and plotted with gnuplot or the like.

The analog program shown in the picture at the beginning is a Hindmarsh-Rose model of a bursting and spiking neuron. [https://analogparadigm.com/downloads/alpaca_28.pdf] Plotting the data captured with the sequence of commands shown above with gnuplot yields a result like this:

[[File:Hindmarsh_rose_x.jpg|center|500px]]

File:Hindmarsh rose x.jpg

2021-10-14T09:15:34Z

Ulmann:

Hybrid Computer

2021-10-14T09:14:24Z

Ulmann:

= Hybrid computers =

A '''hybrid computer''' is a computer consisting of an [[analog computer]] and a digital computer. The two worlds are connected with ADCs/DACs (analog to digital/digital to analog converters).

[[Category:Fundamentals]]

== A simple Arduino based hybrid controller ==

[[File:Hc_setup.jpg|center|500px]]

This section describes a simple yet quite powerful hybrid computer consisting of THE ANALOG THING and an attached Arduino Mega 2650 micro controller board. Since THE ANALOG THING features a dedicated connector for attaching an external digital computer, only a few connections between this HYBRID connector and the Arduino are required to setup the hardware:

{| class="wikitable"
|+ Caption text
|-
! HYBRID pin !! description !! destination on Arduino
|-
| 2 || analog x output || AnalogIn 0
|-
| 4 || analog y output || AnalogIn 1
|-
| 6 || analog z output || AnalogIn 2
|-
| 8 || analog u output || AnalogIn 3
|-
| 9+10 || GND || GND
|-
| 13 || enable hybrid mode || D2
|-
| 14 || MOP || D3
|-
| 16 || MIC || D4
|}

These connections were made using a piggy back board suitable for an Arduino Mega 2650 which only contains a 2x8 pin header and the connections listed in the above table:

[[File:Arduino_hc.jpg|center|500px]]

== Software ==

The required software for this hybrid controller can be found at [https://github.com/anabrid/hardware/blob/main/the-analog-thing/arduino_2650_hybrid_controller/simple_hybrid_controller/simple_hybrid_controller.ino]. It requires two additional libraries, TimerThree and TimerFive, to compile. These two timer libraries are necessary for the accurate timing of operating modes and data logging and must be installed for your Arduino IDE before compiling the source.

== Commands ==

* arm: Arm the data logger. Data logging can be performed during a single run (see below) and will automatically start when the logger has been armed before starting the single run. After completing the single run, the logger is disarmed.
* channels=value: Set the number of analog channels to be logged. The value can range between 1 and 4.
* disable: Disable the hybrid controller. This is the normal startup mode; THE ANALOG THING will work as a standalone analog computer when the hybrid port is disabled.
* enable: Enable the hybrid controller. In this case, THE ANALOG THING is is under control of the attached microcontroller and the mode switch is disabled.
* halt: Put THE ANALOG THING into HALT-mode.
* help: Print a short help text.
* ic: Set THE ANALOG THING to IC-mode.
* ictime=value: Set the IC-time to value milliseconds. This is necessary for any single or repetitive run. Typically the duration of IC-mode is a few milliseconds. (If there are any integrators in an analog program with their time scale factor set to SLOW, the IC-time should be high enough so that these integrators can settle to the selected initial condition.)
* interval=value: Set the sampling interval for the data logger to value milliseconds.
* op: Set THE ANALOG THING to OP-mode.
* optime=value: Set the OP-time to value milliseconds.
* rep: Start repetitive operation. THE ANALOG THING will from now on cycle between IC and OP with the respective times set by the ictime and optime commands. Repetitive operation is ended when the computer is explicitly set into IC-, OP- or HALT-mode.
* run: Start a single run with the IC- and OP-phase duration determined by the times set with the ictime and optime commands. Only during such a single run data logging is possible.
* status: Return status information.

== Typical operation ==
A typical sequence of commands to run an analog program once while collecting
data looks like this:

```
ictime=50
optime=3000
interval=5
channels=1
arm
enable
run
```

This will print the AD converted values during the single run. These can then
be copied to a file (by copy/paste) and plotted with gnuplot or the like.

The analog program shown in the picture at the beginning is a Hindmarsh-Rose model of a bursting and spiking neuron. [https://analogparadigm.com/downloads/alpaca_28.pdf]

Hybrid Computer

2021-10-14T09:10:35Z

Ulmann:

Hybrid Computer

2021-10-14T09:04:51Z

Ulmann:

A '''hybrid computer''' is a computer consisting of an [[analog computer]] and a digital computer. The two worlds are connected with ADCs/DACs (analog to digital/digital to analog converters).

[[Category:Fundamentals]]

[[File:Hc_setup.jpg|center|500px]]

This section describes a simple yet quite powerful hybrid computer consisting of THE ANALOG THING and an attached Arduino Mega 2650 micro controller board. Since THE ANALOG THING features a dedicated connector for attaching an external digital computer, only a few connections between this HYBRID connector and the Arduino are required to setup the hardware:

{| class="wikitable"
|+ Caption text
|-
! HYBRID pin !! description !! destination on Arduino
|-
| 2 || analog x output || AnalogIn 0
|-
| 4 || analog y output || AnalogIn 1
|-
| 6 || analog z output || AnalogIn 2
|-
| 8 || analog u output || AnalogIn 3
|-
| 9+10 || GND || GND
|-
| 13 || enable hybrid mode || D2
|-
| 14 || MOP || D3
|-
| 16 || MIC || D4
|}

These connections were made using a piggy back board suitable for an Arduino Mega 2650 which only contains a 2x8 pin header and the connections listed in the above table:

[[File:Arduino_hc.jpg|center|500px]]

File:Arduino hc.jpg

2021-10-14T09:03:47Z

Ulmann:

File:Hc setup.jpg

2021-10-14T09:02:56Z

Ulmann:

Hybrid Computer

2021-10-14T09:01:22Z

Ulmann:

A '''hybrid computer''' is a computer consisting of an [[analog computer]] and a digital computer. The two worlds are connected with ADCs/DACs (analog to digital/digital to analog converters).

[[Category:Fundamentals]]

This section describes a simple yet quite powerful hybrid computer consisting of THE ANALOG THING and an attached Arduino Mega 2650 micro controller board. Since THE ANALOG THING features a dedicated connector for attaching an external digital computer, only a few connections between this HYBRID connector and the Arduino are required to setup the hardware:

{| class="wikitable"
|+ Caption text
|-
! HYBRID pin !! description !! destination on Arduino
|-
| 2 || analog x output || AnalogIn 0
|-
| 4 || analog y output || AnalogIn 1
|-
| 6 || analog z output || AnalogIn 2
|-
| 8 || analog u output || AnalogIn 3
|-
| 9+10 || GND || GND
|-
| 13 || enable hybrid mode || D2
|-
| 14 || MOP || D3
|-
| 16 || MIC || D4
|}

Comparator

2021-09-26T10:50:43Z

Ulmann:

[[File:Comparator.png|thumb|The six different jacks of a single Comparator]]

The '''Comparator''' is a logic element which puts different signals on the output, depending on the input. [[The Analog Thing]] features two comparators.

Given two inputs <code>A</code> and <code>B</code>, the comparator tests whether the sum of the two input values is larger or smaller zero. If it is larger, the input <code>>0</code> is directed to <code>OUT</code>, if smaller, the input <code><0</code> is directed to <code>OUT</code>.

Technically, the comparators of THAT are implemented with Schmitt Triggers and two analog switches. See [[:File:Anathing_v1.0_base_4.pdf]] for the schematics.

The schematic is not incomplete - the input resistor networks are located on the front panel (as for the summers, inverters, and integrators).

[[Category:Components of The Analog Thing]]

Comparator

2021-09-26T10:50:10Z

Ulmann:

[[File:Comparator.png|thumb|The six different jacks of a single Comparator]]

The '''Comparator''' is a logic element which puts different signals on the output, depending on the input. [[The Analog Thing]] features two comparators.

Given two inputs <code>A</code> and <code>B</code>, the comparator tests {{fill}} (A? B?) whether the input value is larger or smaller zero. If it is larger, the input <code>>0</code> is directed to <code>OUT</code>, if smaller, the input <code><0</code> is directed to <code>OUT</code>.

Technically, the comparators of THAT are implemented with Schmitt Triggers and two analog switches. See [[:File:Anathing_v1.0_base_4.pdf]] for the schematics.

The schematic is not incomplete - the input resistor networks are located on the front panel (as for the summers, inverters, and integrators).

[[Category:Components of The Analog Thing]]

Talk:Multiplier

2021-09-26T10:49:05Z

Ulmann:

Why are the inputs of the multipliers divided by 2 and the output multiplied by 4?

The AD633 does not feature rail-to-rail inputs. This in conjunction with the lower supply voltage used here necessitates a scale-down of the input values with a subsequent amplifier stage.

The Analog Thing

2021-09-20T15:59:55Z

Ulmann:

[[File:Cta bg.jpg|thumb|A closeup of a THAT in [[Minion]] mode with [[Oscilloscope]] probes attached.]]
'''The Analog Thing''' (abbreviated as ''THAT'') is a high-quality, low-cost, open-source, and not-for-profit cutting-edge analog computer analog computer developed by [[anabrid]] under the brand Analog Paradigm for educational and recreational purposes. THAT will be available for sale at a net cost price of around €300. The component list, circuit diagrams and circuit board layouts of THAT are all Open Source. Information on how to build THAT from scratch and how to operate it can be found on this wiki.

== Version 1.0: Specifications and Photos ==
The prototype consists of two main parts: A top PCB with gold-plated through-hole sockets suitable for
2 mm patch cables, and a base PCB, which contains the main electronic circuitry, the
manual controls and ports for external connectivity. A [[Panel Meter]] mounted in the top PCB can be used to display coefficient values and, when the device is set to the repetitive operation mode, its operation time.

The size of the analog computer is 203mm x 240mm x 35 mm (with knobs).

<gallery>
File:Promo 1.jpg|A first glance
File:Front Board v1.0.jpg|Front board v1.0
File:Base Board v1.0 patched.jpg|Base board v1.0
File:Test Circuit closeup v1.0.jpg|Close-up of a test circuit
</gallery>

== Schematics ==
The schematics for the base board are [[Open Source]] and can be found here:

<gallery>
File:anathing_v1.0_base_1.pdf|BASE part 1
File:anathing_v1.0_base_2.pdf|BASE part 2
File:anathing_v1.0_base_3.pdf|BASE part 3
File:anathing_v1.0_base_4.pdf|BASE part 4
</gallery>

The schematic of the FRONT PCB can be found here:

<gallery>
File:anathing_v1.0_front.pdf|FRONT
</gallery>

== Instructions and further reading ==

* [[Assembly instructions]]
* [[Testing]] instructions
* [[Components of The Analog Thing]]
* [[The Analog Thing FAQ]]

[[Category:Manual]]

Applications

2021-09-01T16:47:35Z

Ulmann:

== List of examples ==
Here are some application examples for THAT.

<gallery>
File:lorenz_attractor_small.jpg|400px|link=Lorenz|[[Lorenz]]
File:SEIR_result_2_small.jpg|400px|link=SEIR|[[SEIR]]
File:hindmarsh_rose_50ms_div.jpg|400px|link=Hindmarsh-Rose neuron model|[[Hindmarsh-Rose neuron model]]
</gallery>

== Other sources ==

Most of the Analog Paradigm application notes (https://analogparadigm.com/documentation.html or https://github.com/anabrid/Model-1/tree/main/application_notes) are also applicable for THAT.

[[Category:Applications]]

Applications

2021-09-01T16:47:17Z

Ulmann:

This page should list some '''Circuits/Applications''' for [[The Analog Thing]].

== List of examples ==
Here are some application examples for THAT.

<gallery>
File:lorenz_attractor_small.jpg|400px|link=Lorenz|[[Lorenz]]
File:SEIR_result_2_small.jpg|400px|link=SEIR|[[SEIR]]
File:hindmarsh_rose_50ms_div.jpg|400px|link=Hindmarsh-Rose neuron model|[[Hindmarsh-Rose neuron model]]
</gallery>

== Other sources ==

Most of the Analog Paradigm application notes (https://analogparadigm.com/documentation.html or https://github.com/anabrid/Model-1/tree/main/application_notes) are also applicable for THAT.

[[Category:Applications]]

Applications

2021-09-01T16:46:21Z

Ulmann:

This page should list some '''Circuits/Applications''' for [[The Analog Thing]].

== List of examples ==
* Most of the Analog Paradigm application notes (https://analogparadigm.com/documentation.html or https://github.com/anabrid/Model-1/tree/main/application_notes) are also applicable for THAT.

<gallery>
File:lorenz_attractor_small.jpg|400px|link=Lorenz|[[Lorenz]]
File:SEIR_result_2_small.jpg|400px|link=SEIR|[[SEIR]]
File:hindmarsh_rose_50ms_div.jpg|400px|link=Hindmarsh-Rose neuron model|[[Hindmarsh-Rose neuron model]]
</gallery>

[[Category:Applications]]

Lorenz

2021-09-01T13:25:25Z

Ulmann:

In 1963 Edward Norton Lorenz (23.05.1917 - 16.04.2008) developed a model for atmospheric convection (see https://journals.ametsoc.org/view/journals/atsc/20/2/1520-0469_1963_020_0130_dnf_2_0_co_2.xml and https://www.math.uni-hamburg.de/home/lauterbach/scripts/seminar03/prill.pdf for more details). This model, which was simulated on a tiny digital computer (Royal McBee LPG-30), showed an interesting behaviour which gave rise to research on chaotic attractors.

The system itself is described by three coupled differential equations (x' denotes the first derivative of x with respect to time here - more typically it would be written with a dot over the variable name):

x'=σ(y-x) 
y'=x(ρ-z)-y 
z'=xy-βz 

In the original system the parameters are σ=10 (that is the Prandtl number), β=8/3 (this represents the cell geometry underlying the model), and ρ=28 (relative Rayleigh number). Obviously, this cannot be implemented on an analog computer directly as it is not scaled to the interval [-1,1]. Scaling this system is a bit cumbersome: First, the domains of all involved variables have to be determined. Scaling and rewriting the system yields the following set of equations:

x=∫1.8y-x dt+C 
s=1-2.678z 
y=∫1.5556xs-0.1y dt 
z=∫1.5xy-0.2667z dt 

The constant C basically represents the initial condition of that integrator and is not critical. Since each integrator and summer performs an implicit change of sign in an analog computer such as THAT the above equations can be simplified a bit:

-x=-∫1.8y-x dt+C 
-z=-∫1.5xy-0.2667z dt 
s=-(1-2.68z) 
r=-xs 
-y=-∫1.536r-0.1y dt 

These equations can now be implemented directly on an analog computer as shown in this program:

[[File:lorenz_program.jpg|768px]]

A typical setup looks like this:

[[File:lorenz_THAT_setup.jpg|768px]]

Using a classic analog oscilloscope in xy-mode and a camera capable of long term exposures, the beauty of this particular chaotic attractor can be captured. Shown in the following is the phase space plot of the variables -x and s:

[[File:lorenz_attractor_long_term_exposure.jpg|768px]]

Since THAT offers two time constants for its integrators it is possible to slow down the computation by a factor of 100 by connecting the SLOW jack of every integrator to the output jack of the same integrator. Thus it is possible to control a classic plotter:

[[File:Lorenz_plotter.mp4]]

Lorenz

2021-09-01T13:23:32Z

Ulmann:

In 1963 Edward Norton Lorenz (23.05.1917 - 16.04.2008) developed a model for atmospheric convection (see https://journals.ametsoc.org/view/journals/atsc/20/2/1520-0469_1963_020_0130_dnf_2_0_co_2.xml and https://www.math.uni-hamburg.de/home/lauterbach/scripts/seminar03/prill.pdf for more details). This model, which was simulated on a tiny digital computer (Royal McBee LPG-30), showed an interesting behaviour which gave rise to research on chaotic attractors.

The system itself is described by three coupled differential equations (x' denotes the first derivative of x with respect to time here - more typically it would be written with a dot over the variable name):

x'=σ(y-x) 
y'=x(ρ-z)-y 
z'=xy-βz 

In the original system the parameters are σ=10 (that is the Prandtl number), β=8/3 (this represents the cell geometry underlying the model), and ρ=28 (relative Rayleigh number). Obviously, this cannot be implemented on an analog computer directly as it is not scaled to the interval [-1,1]. Scaling this system is a bit cumbersome: First, the domains of all involved variables have to be determined. Scaling and rewriting the system yields the following set of equations:

x=∫1.8y-x dt+C 
s=1-2.678z 
y=∫1.5556xs-0.1y dt 
z=∫1.5xy-0.2667z dt 

The constant C basically represents the initial condition of that integrator and is not critical. Since each integrator and summer performs an implicit change of sign in an analog computer such as THAT the above equations can be simplified a bit:

-x=-∫1.8y-x dt+C 
-z=-∫1.5xy-0.2667z dt 
s=-(1-2.68z) 
r=-xs 
-y=-∫1.536r-0.1y dt 

These equations can now be implemented directly on an analog computer as shown in this program:

[[File:lorenz_program.jpg|768px]]

A typical setup looks like this:

[[File:lorenz_THAT_setup.jpg|768px]]

Using a classic analog oscilloscope in xy-mode and a camera capable of long term exposures, the beauty of this particular chaotic attractor can be captured. Shown in the following is the phase space plot of the variables -x and -z:

[[File:lorenz_attractor_long_term_exposure.jpg|768px]]

Since THAT offers two time constants for its integrators it is possible to slow down the computation by a factor of 100 by connecting the SLOW jack of every integrator to the output jack of the same integrator. Thus it is possible to control a classic plotter:

[[File:Lorenz_plotter.mp4]]

File:Lorenz plotter.mp4

2021-09-01T13:22:09Z

Ulmann:

Lorenz

2021-09-01T12:01:40Z

Ulmann:

In 1963 Edward Norton Lorenz (23.05.1917 - 16.04.2008) developed a model for atmospheric convection (see https://journals.ametsoc.org/view/journals/atsc/20/2/1520-0469_1963_020_0130_dnf_2_0_co_2.xml and https://www.math.uni-hamburg.de/home/lauterbach/scripts/seminar03/prill.pdf for more details). This model, which was simulated on a tiny digital computer (Royal McBee LPG-30), showed an interesting behaviour which gave rise to research on chaotic attractors.

The system itself is described by three coupled differential equations (x' denotes the first derivative of x with respect to time here - more typically it would be written with a dot over the variable name):

x'=σ(y-x) 
y'=x(ρ-z)-y 
z'=xy-βz 

In the original system the parameters are σ=10 (that is the Prandtl number), β=8/3 (this represents the cell geometry underlying the model), and ρ=28 (relative Rayleigh number). Obviously, this cannot be implemented on an analog computer directly as it is not scaled to the interval [-1,1]. Scaling this system is a bit cumbersome: First, the domains of all involved variables have to be determined. Scaling and rewriting the system yields the following set of equations:

x=∫1.8y-x dt+C 
s=1-2.678z 
y=∫1.5556xs-0.1y dt 
z=∫1.5xy-0.2667z dt 

The constant C basically represents the initial condition of that integrator and is not critical. Since each integrator and summer performs an implicit change of sign in an analog computer such as THAT the above equations can be simplified a bit:

-x=-∫1.8y-x dt+C 
-z=-∫1.5xy-0.2667z dt 
s=-(1-2.68z) 
r=-xs 
-y=-∫1.536r-0.1y dt 

These equations can now be implemented directly on an analog computer as shown in this program:

[[File:lorenz_program.jpg|768px]]

A typical setup looks like this:

[[File:lorenz_THAT_setup.jpg|768px]]

Using a classic analog oscilloscope in xy-mode and a camera capable of long term exposures, the beauty of this particular chaotic attractor can be captured. Shown in the following is the phase space plot of the variables -x and -z:

[[File:lorenz_attractor_long_term_exposure.jpg|768px]]

Since THAT offers two time constants for its integrators it is possible to slow down the computation by a factor of 100 by connecting the SLOW jack of every integrator to the output jack of the same integrator. Thus it is possible to control a classic plotter:

[[File:Lorenz_plotter.mov]]

File:Lorenz plotter.mov

2021-09-01T11:59:45Z

Ulmann:

Lorenz

2021-09-01T11:56:11Z

Ulmann:

In 1963 Edward Norton Lorenz (23.05.1917 - 16.04.2008) developed a model for atmospheric convection (see https://journals.ametsoc.org/view/journals/atsc/20/2/1520-0469_1963_020_0130_dnf_2_0_co_2.xml and https://www.math.uni-hamburg.de/home/lauterbach/scripts/seminar03/prill.pdf for more details). This model, which was simulated on a tiny digital computer (Royal McBee LPG-30), showed an interesting behaviour which gave rise to research on chaotic attractors.

The system itself is described by three coupled differential equations (x' denotes the first derivative of x with respect to time here - more typically it would be written with a dot over the variable name):

x'=σ(y-x) 
y'=x(ρ-z)-y 
z'=xy-βz 

In the original system the parameters are σ=10 (that is the Prandtl number), β=8/3 (this represents the cell geometry underlying the model), and ρ=28 (relative Rayleigh number). Obviously, this cannot be implemented on an analog computer directly as it is not scaled to the interval [-1,1]. Scaling this system is a bit cumbersome: First, the domains of all involved variables have to be determined. Scaling and rewriting the system yields the following set of equations:

x=∫1.8y-x dt+C 
s=1-2.678z 
y=∫1.5556xs-0.1y dt 
z=∫1.5xy-0.2667z dt 

The constant C basically represents the initial condition of that integrator and is not critical. Since each integrator and summer performs an implicit change of sign in an analog computer such as THAT the above equations can be simplified a bit:

-x=-∫1.8y-x dt+C 
-z=-∫1.5xy-0.2667z dt 
s=-(1-2.68z) 
r=-xs 
-y=-∫1.536r-0.1y dt 

These equations can now be implemented directly on an analog computer as shown in this program:

[[File:lorenz_program.jpg|768px]]

A typical setup looks like this:

[[File:lorenz_THAT_setup.jpg|768px]]

Using a classic analog oscilloscope in xy-mode and a camera capable of long term exposures, the beauty of this particular chaotic attractor can be captured. Shown in the following is the phase space plot of the variables -x and -z:

[[File:lorenz_attractor_long_term_exposure.jpg|768px]]

File:Lorenz plotter.jpg

2021-09-01T11:55:59Z

Ulmann: Plotter connected to THAT plotting a Lorenz attractor.

== Summary ==
Plotter connected to THAT plotting a Lorenz attractor.

Lorenz

2021-09-01T11:11:25Z

Ulmann:

In 1963 Edward Norton Lorenz (23.05.1917 - 16.04.2008) developed a model for atmospheric convection (see https://journals.ametsoc.org/view/journals/atsc/20/2/1520-0469_1963_020_0130_dnf_2_0_co_2.xml and https://www.math.uni-hamburg.de/home/lauterbach/scripts/seminar03/prill.pdf for more details). This model, which was simulated on a tiny digital computer (Royal McBee LPG-30), showed an interesting behaviour which gave rise to research on chaotic attractors.

The system itself is described by three couple differential equations (x' denotes the first derivative of x with respect to time here - more typically it would be written with a dot over the variable name):

x'=σ(y-x) 
y'=x(ρ-z)-y 
z'=xy-βz 

In the original system the parameters are σ=10 (that is the Prandtl number), β=8/3 (this represents the cell geometry underlying the model), and ρ=28 (relative Rayleigh number). Obviously, this cannot be implemented on an analog computer directly as it is not scaled to the interval [-1,1]. Scaling this system is a bit cumbersome: First, the domains of all involved variables have to be determined. Scaling and rewriting the system yields the following set of equations:

x=∫1.8y-x dt+C 
s=1-2.678z 
y=∫1.5556xs-0.1y dt 
z=∫1.5xy-0.2667z dt 

The constant C basically represents the initial condition of that integrator and is not critical. Since each integrator and summer performs an implicit change of sign in an analog computer such as THAT the above equations can be simplified a bit:

-x=-∫1.8y-x dt+C 
-z=-∫1.5xy-0.2667z dt 
s=-(1-2.68z) 
r=-xs 
-y=-∫1.536r-0.1y dt 

These equations can now be implemented directly on an analog computer as shown in this program:

[[File:lorenz_program.jpg|768px]]

A typical setup looks like this:

[[File:lorenz_THAT_setup.jpg|768px]]

Using a classic analog oscilloscope in xy-mode and a camera capable of long term exposures, the beauty of this particular chaotic attractor can be captured. Shown in the following is the phase space plot of the variables -x and -z:

[[File:lorenz_attractor_long_term_exposure.jpg|768px]]

Lorenz

2021-09-01T10:54:02Z

Ulmann:

In 1963 Edward Norton Lorenz (23.05.1917 - 16.04.2008) developed a model for atmospheric convection (see https://journals.ametsoc.org/view/journals/atsc/20/2/1520-0469_1963_020_0130_dnf_2_0_co_2.xml and https://www.math.uni-hamburg.de/home/lauterbach/scripts/seminar03/prill.pdf for more details). This model, which was simulated on a tiny digital computer (Royal McBee LPG-30), showed an interesting behaviour which gave rise to research on chaotic attractors.

The system itself is described by three couple differential equations (x' denotes the first derivative of x with respect to time here - more typically it would be written with a dot over the variable name):

x'=σ(y-x) 
y'=x(ρ-z)-y 
z'=xy-βz 

In the original system the parameters are σ=10, β=8/3, and ρ=28. Obviously, this cannot be implemented on an analog computer directly as it is not scaled to the interval [-1,1]. Scaling this system is a bit cumbersome: First, the domains of all involved variables have to be determined. Scaling and rewriting the system yields the following set of equations:

x=∫1.8y-x dt+C 
s=1-2.678z 
y=∫1.5556xs-0.1y dt 
z=∫1.5xy-0.2667z dt 

The constant C basically represents the initial condition of that integrator and is not critical. Since each integrator and summer performs an implicit change of sign in an analog computer such as THAT the above equations can be simplified a bit:

-x=-∫1.8y-x dt+C 
-z=-∫1.5xy-0.2667z dt 
s=-(1-2.68z) 
r=-xs 
-y=-∫1.536r-0.1y dt 

These equations can now be implemented directly on an analog computer as shown in this program:

[[File:lorenz_program.jpg|768px]]

A typical setup looks like this:

[[File:lorenz_THAT_setup.jpg|768px]]

Using a classic analog oscilloscope in xy-mode and a camera capable of long term exposures, the beauty of this particular chaotic attractor can be captured. Shown in the following is the phase space plot of the variables -x and -z:

[[File:lorenz_attractor_long_term_exposure.jpg|768px]]

File:Lorenz attractor long term exposure.jpg

2021-09-01T10:51:27Z

Ulmann: Long term exposure picture of a Lorenz attractor.

== Summary ==
Long term exposure picture of a Lorenz attractor.

Lorenz

2021-09-01T10:51:21Z

Ulmann:

In 1963 Edward Norton Lorenz (23.05.1917 - 16.04.2008) developed a model for atmospheric convection (see https://journals.ametsoc.org/view/journals/atsc/20/2/1520-0469_1963_020_0130_dnf_2_0_co_2.xml and https://www.math.uni-hamburg.de/home/lauterbach/scripts/seminar03/prill.pdf for more details). This model, which was simulated on a tiny digital computer (Royal McBee LPG-30), showed an interesting behaviour which gave rise to research on chaotic attractors.

The system itself is described by three couple differential equations (x' denotes the first derivative of x with respect to time here - more typically it would be written with a dot over the variable name):

x'=σ(y-x) 
y'=x(ρ-z)-y 
z'=xy-βz 

In the original system the parameters are σ=10, β=8/3, and ρ=28. Obviously, this cannot be implemented on an analog computer directly as it is not scaled to the interval [-1,1]. Scaling this system is a bit cumbersome: First, the domains of all involved variables have to be determined. Scaling and rewriting the system yields the following set of equations:

x=∫1.8y-x dt+C 
s=1-2.678z 
y=∫1.5556xs-0.1y dt 
z=∫1.5xy-0.2667z dt 

The constant C basically represents the initial condition of that integrator and is not critical. Since each integrator and summer performs an implicit change of sign in an analog computer such as THAT the above equations can be simplified a bit:

-x=-∫1.8y-x dt+C 
-z=-∫1.5xy-0.2667z dt 
s=-(1-2.68z) 
r=-xs 
-y=-∫1.536r-0.1y dt 

These equations can now be implemented directly on an analog computer as shown in this program:

[[File:lorenz_program.jpg|768px]]

A typical setup looks like this:

[[File:lorenz_THAT_setup.jpg|768px]]

Using a classic analog oscilloscope in xy-mode and a camera capable of long term exposures, the beauty of this particular chaotic attractor can be captured:

[[File:lorenz_attractor_long_term_exposure.jpg|768px]]

Lorenz

2021-09-01T10:48:46Z

Ulmann:

In 1963 Edward Norton Lorenz (23.05.1917 - 16.04.2008) developed a model for atmospheric convection (see https://journals.ametsoc.org/view/journals/atsc/20/2/1520-0469_1963_020_0130_dnf_2_0_co_2.xml and https://www.math.uni-hamburg.de/home/lauterbach/scripts/seminar03/prill.pdf for more details). This model, which was simulated on a tiny digital computer (Royal McBee LPG-30), showed an interesting behaviour which gave rise to research on chaotic attractors.

The system itself is described by three couple differential equations (x' denotes the first derivative of x with respect to time here - more typically it would be written with a dot over the variable name):

x'=σ(y-x) 
y'=x(ρ-z)-y 
z'=xy-βz 

In the original system the parameters are σ=10, β=8/3, and ρ=28. Obviously, this cannot be implemented on an analog computer directly as it is not scaled to the interval [-1,1]. Scaling this system is a bit cumbersome: First, the domains of all involved variables have to be determined. Scaling and rewriting the system yields the following set of equations:

x=∫1.8y-x dt+C 
s=1-2.678z 
y=∫1.5556xs-0.1y dt 
z=∫1.5xy-0.2667z dt 

The constant C basically represents the initial condition of that integrator and is not critical. Since each integrator and summer performs an implicit change of sign in an analog computer such as THAT the above equations can be simplified a bit:

-x=-∫1.8y-x dt+C 
-z=-∫1.5xy-0.2667z dt 
s=-(1-2.68z) 
r=-xs 
-y=-∫1.536r-0.1y dt 

These equations can now be implemented directly on an analog computer as shown in this program:

[[File:lorenz_program.jpg|768px]]

A typical setup looks like this:

[[File:lorenz_THAT_setup.jpg|768px]]

Lorenz

2021-09-01T10:48:29Z

Ulmann:

In 1963 Edward Norton Lorenz (23.05.1917 - 16.04.2008) developed a model for atmospheric convection (see https://journals.ametsoc.org/view/journals/atsc/20/2/1520-0469_1963_020_0130_dnf_2_0_co_2.xml and https://www.math.uni-hamburg.de/home/lauterbach/scripts/seminar03/prill.pdf for more details). This model, which was simulated on a tiny digital computer (Royal McBee LPG-30), showed an interesting behaviour which gave rise to research on chaotic attractors.

The system itself is described by three couple differential equations (x' denotes the first derivative of x with respect to time here - more typically it would be written with a dot over the variable name):

x'=σ(y-x) 
y'=x(ρ-z)-y 
z'=xy-βz 

In the original system the parameters are σ=10, β=8/3, and ρ=28. Obviously, this cannot be implemented on an analog computer directly as it is not scaled to the interval [-1,1]. Scaling this system is a bit cumbersome: First, the domains of all involved variables have to be determined. Scaling and rewriting the system yields the following set of equations:

x=∫1.8y-x dt+C 
s=1-2.678z 
y=∫1.5556xs-0.1y dt 
z=∫1.5xy-0.2667z dt 

The constant C basically represents the initial condition of that integrator and is not critical. Since each integrator and summer performs an implicit change of sign in an analog computer such as THAT the above equations can be simplified a bit:

-x=-∫1.8y-x dt+C 
-z=-∫1.5xy-0.2667z dt 
s=-(1-2.68z) 
r=-xs 
-y=-∫1.536r-0.1y dt 

These equations can now be implemented directly on an analog computer as shown in this program:
[[File:lorenz_program.jpg|768px]]

A typical setup looks like this:
[[File:lorenz_THAT_setup.jpg|768px]]

File:Lorenz THAT setup.jpg

2021-09-01T10:47:41Z

Ulmann:

Lorenz

2021-09-01T09:57:47Z

Ulmann:

File:Lorenz program.jpg

2021-09-01T09:57:04Z

Ulmann: Program for the Lorenz attractor.

== Summary ==
Program for the Lorenz attractor.

Lorenz

2021-09-01T09:55:25Z

Ulmann:

Lorenz

2021-09-01T09:50:05Z

Ulmann:

Lorenz

2021-09-01T09:49:39Z

Ulmann:

Lorenz

2021-09-01T09:47:50Z

Ulmann: Created page with "In 1963 Edward Norton Lorenz (23.05.1917 - 16.04.2008) developed a model for atmospheric convection (see https://journals.ametsoc.org/view/journals/atsc/20/2/1520-0469_1963_02..."

Applications

2021-09-01T09:44:52Z

Ulmann:

This page should list some '''Circuits/Applications''' for [[The Analog Thing]].

== List of examples ==
* Most of the Analog Paradigm application notes (https://analogparadigm.com/documentation.html or https://github.com/anabrid/Model-1/tree/main/application_notes) are also applicable for THAT.

<gallery>
File:lorenz_attractor_small.jpg|400px|link=lorenz|[[lorenz]]
File:SEIR_result_2_small.jpg|400px|link=SEIR|[[SEIR]]
File:hindmarsh_rose_50ms_div.jpg|400px|link=Hindmarsh-Rose neuron model|[[Hindmarsh-Rose neuron model]]
</gallery>

[[Category:Applications]]

Applications

2021-09-01T09:42:57Z

Ulmann:

This page should list some '''Circuits/Applications''' for [[The Analog Thing]].

== List of examples ==
* Most of the Analog Paradigm application notes (https://analogparadigm.com/documentation.html or https://github.com/anabrid/Model-1/tree/main/application_notes are also applicable for the THAT.

<gallery>
File:lorenz_attractor_small.jpg|400px|link=lorenz|[[lorenz]]
File:SEIR_result_2_small.jpg|400px|link=SEIR|[[SEIR]]
File:hindmarsh_rose_50ms_div.jpg|400px|link=Hindmarsh-Rose neuron model|[[Hindmarsh-Rose neuron model]]
</gallery>

[[Category:Applications]]

File:Lorenz attractor small.jpg

2021-09-01T09:42:13Z

Ulmann: Small picture of a typical Lorenz attractor display

== Summary ==
Small picture of a typical Lorenz attractor display

File:SEIR result 2 small.jpg

2021-08-20T12:16:44Z

Ulmann: Ulmann uploaded a new version of File:SEIR result 2 small.jpg

== Summary ==
Typical result of a SEIR simulation.

Hindmarsh-Rose neuron model

2021-08-20T12:10:10Z

Ulmann:

Beginning in the 20th centure, the behaviour of neurons has been described by mathematical models, beginning with Louis Lapicque's "integrate and fire" model in 1907. In the 1960s Richard FitzHugh and J. Nagumo developed a more sophisticated model. In the early 1980s, Hindmarsh and Rose published an even more realistic model of a spiking and burstin neuron, the Hindmarsh-Rose model.

This is desribed as a system of three coupled differential equations:

[[File:hindmarsh_rose_equations.jpg|256px]]

with the parameter a=1, b=3, c=1, d=5, r=0.001, s=4 and xr=-8/5 and initial conditions of 2 for all three equations. This system of equations and parameters must be, of course, scaled for implementation on an analog computer. The resulting program looks like this:

[[File:Hindmarsh_rose_prg.pdf|768px]]

The output of the simulated neuron is x, while Iext represents the input to the neuron. Setting Iext to +1 will cause the neuron to burst with sequences of spikes as shown in this actual screen shot from a digital oscilloscope:

[[File:Hindmarsh_rose_result.jpg|768px]]

The actual implementation of this program looks like this:

[[File:Hindmarsh_rose_program.jpg|512px]]

File:Hindmarsh rose program.jpg

2021-08-20T12:09:12Z

Ulmann: Ulmann uploaded a new version of File:Hindmarsh rose program.jpg

Hindmarsh-Rose neuron model

2021-08-20T12:07:46Z

Ulmann:

File:Hindmarsh rose program.jpg

2021-08-20T12:07:37Z

Ulmann:

Hindmarsh-Rose neuron model

2021-08-20T12:06:57Z

Ulmann:

File:Hindmarsh rose result.jpg

2021-08-20T12:06:46Z

Ulmann:

File:Hindmarsh rose prg.pdf

2021-08-20T12:06:03Z

Ulmann: Ulmann uploaded a new version of File:Hindmarsh rose prg.pdf

Hindmarsh-Rose neuron model

2021-08-20T12:04:05Z

Ulmann:

Beginning in the 20th centure, the behaviour of neurons has been described by mathematical models, beginning with Louis Lapicque's "integrate and fire" model in 1907. In the 1960s Richard FitzHugh and J. Nagumo developed a more sophisticated model. In the early 1980s, Hindmarsh and Rose published an even more realistic model of a spiking and burstin neuron, the Hindmarsh-Rose model.

This is desribed as a system of three coupled differential equations:

[[File:hindmarsh_rose_equations.jpg|256px]]

with the parameter a=1, b=3, c=1, d=5, r=0.001, s=4 and xr=-8/5 and initial conditions of 2 for all three equations. This system of equations and parameters must be, of course, scaled for implementation on an analog computer. The resulting program looks like this:

[[File:hindmarsh_rose_prg.pdf|768px]]

The output of the simulated neuron is x, while Iext represents the input to the neuron. Setting Iext to +1 will cause the neuron to burst with sequences of spikes as shown in this actual screen shot from a digital oscilloscope:

Hindmarsh-Rose neuron model

2021-08-20T12:02:35Z

Ulmann: