**Biologically Inspired**:- The AdEx model is designed to capture essential features of real neurons.
- It incorporates
**biologically realistic time constants**and mimics neuronal behavior, including**refractory periods**and**spike-frequency adaptation (SFA)**.

**Model Description**:- The AdEx model consists of two variables: membrane potential ((V)) and adaptation variable ((w)).
- The first equation describes the dynamics of the membrane potential, including an activation term with an
**exponential voltage dependence**. - Voltage is coupled to a second equation that describes adaptation.
- Both variables are
**reset**if an action potential is triggered.

**Firing Patterns**:- The AdEx model is capable of describing various
**neuronal firing patterns**, including:**Adapting**: Firing rate decreases over time.**Bursting**: Rapid sequences of spikes.**Delayed spike initiation**: Delayed response to input.**Initial bursting**: Bursting at the start of stimulation.**Fast spiking**: High-frequency firing.**Regular spiking**: Regular, periodic firing.

- The AdEx model is capable of describing various
**Mathematical Formulatio**n

**Equations**:- Equation (1) describes the membrane potential dynamics:
- The rate of change of membrane potential
**dVmem/dt**is influenced by external, synaptic, and inter-compartment input currents. - It accounts for leak and exponential currents.
**Cmem**represents the membrane capacitance.**Vmem**is the membrane potential.**gleak**and**Vleak**denote the leak conductance and leak potential.**a**represents subthreshold adaptation.**VT**is the exponential threshold, and**∆T**is its slope factor.

- The rate of change of membrane potential
- Equation (2) describes the adaptation current dynamics:
- The rate of change of adaptation current
**dw/dt**depends on the difference between**Vmem**and**Vleak**. - The adaptation variable
**w**is updated by a current**b**at spike time.

- The rate of change of adaptation current

- Equation (1) describes the membrane potential dynamics:
**Hardware Realization**:- For hardware implementation, the adaptation term is transformed.
- The adaptation output current
**w**is expressed as**w = a(Vw – Vleak)**. - This substitution modifies Equation (2) to:
**dVw/dt = gw(Vw – Vmem)**.

**Dendritic Structure**:- The circuit implements controlled conductance between adjacent isopotential membranes.
- The inter-compartmental current flowing through the tunable conductance connecting two membranes is given by:
**Iic = gic(Vi – Vj)**, where**gic**is the inter-compartmental conductance, and**Vi**and**Vj**are potentials of the shunted membranes.

**Energy Efficiency**:- The AdEx model’s computational affordability arises from its
**subthreshold operation**(all transistors operate in the subthreshold region). - It ensures
**extremely low power consumption**, making it suitable for energy-efficient neural circuits.

- The AdEx model’s computational affordability arises from its
**Historical Context**:- Introduced by Brette and Gerstner in
**2005**, the AdEx model builds on features of the exponential integrate-and-fire model and Izhikevich’s 2-variable model. - Read more https://ieeexplore.ieee.org/abstract/document/8419063

- Introduced by Brette and Gerstner in
- About the Circuit

**Purpose and Functionality**:- The adaptation circuit serves to implement
**accelerating and decelerating spike-triggered adaptation**. - It also handles
**adaptation current**based on equations (Eq. 3 and Eq. 4).

- The adaptation circuit serves to implement
**Circuit Overview**:- The circuit is inspired by a first-generation design.
- A simplified schematic is depicted in Fig. 2b.

**Top Right Part of the Circuit (Eq. 3)**:- The output current
**w**is generated by an**OTA (Operational Transconductance Amplifier)**with conductance**ga**. - It emulates the model’s subthreshold conductance parameter
**a**. - The OTA senses the difference between
**Vleak**and**Vw**at its inputs. - A configuration bit
**enVa**switches them to realize negative**ga**.

- The output current
**Lower Right Part of the Schematic (Eq. 4)**:- The input current
**gw(Vw – Vmem)**is integrated on the adaptation capacitor**Cw**. - The membrane is buffered by an OTA labeled
**Aw**. - A tunable floating conductance
**gw**connects the buffered membrane to capacitor**Cw**.

- The input current
**Circuit Activation and Capacitor Merging**:- The circuit is enabled by asserting the bit
**enadapt**and dis-asserting**encapMerge**. - This connects the output to the membrane
**Vmem**via switch**Sa0**while connecting**Cw**to the node**Vw**. - When
**encapMerge**is disabled (e.g., in LIF mode), the unused adaptation capacitor can be merged with the membrane capacitor**Cmem**, effectively increasing the maximum**Cmem**from 2.36 pF to 4.36 pF.

- The circuit is enabled by asserting the bit
**Adaptation Variable and Time Constant**:- The voltage on the capacitor
**Vw**emulates the adaptation variable in the model. - A tunable conductance implements the adaptation time constant
**τw = RwCw**, where**Rw = 1/gw**.

- The voltage on the capacitor
**Charge Pump and Triggering**:- The left side of the circuit features a charge pump.
- An on-chip bias current
**Iw**is sourced or sunk from node**Vw**via pass transistors**Sp**and**Sn**(assuming**enadapt = 1**). - The circuit is triggered by the input event
**fireadapt**, which indicates a digital spike event.

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