Synaptic static is a spell with an interesting effect because it goes into detail about what happens when the spell ends. There are a few other spells that also do this (levitate, haste) but in this case, it has a duration of instantaneous. That puts it into a separate category. Here's the part that is interesting to me: After a failed save, a target has muddled thoughts for 1 minute. ... The target can make an Intelligence saving throw at the end of each of its turns, ending the effect on itself on a success.

So what does this mean? The spell has ended, but an effect lingers for up to a minute. Is the lingering effect of the spell considered part of the spell? Is it magical? On the one hand, the effect is part of the spell's description, and that generally means it's a magical effect. On the other hand, we know that the lingering effects created by other instantaneous spells (animate dead for instance) are not considered magical. It raises a few questions about how this would work in gameplay.

• Can we apply the standard from SAC about animate dead to this circumstance?
• Would a gnome have advantage due to their gnome cunning on subsequent INT saves to end the lingering effect?
• If a creature walked into an anti-magic field, would it suppress the muddled thoughts?

How much of it is magical?

The short answer… all of it.

Ehhh.... I'd probably treat it as a non-magical "curse" for the purpose of whether to use Dispel Magic or Greater Restoration.

Kind of like getting into someone's brain with a magical stick and giving it a little stir. The effect can only be achieved through magic, but the aftereffect is otherwise mundane.

There's multiple things that result in lasting magical effects even after the spell that caused the effects has ended. Find Familiar, is a great example. The familiar is magical and your bond is magical and you have the magical ability to store it and this is all caused by a spell. Yet, it is no longer a spell. It cannot be dispelled. 

Likewise Synaptic Static leaves a lasting magical effect in the mind of your target, but the spell that caused it has ended so this effect cannot be dispelled. 

Dispel Magic only works on active spells. You may target "magical effects" but that is only for targeting. Dispel only ends spells on that target. "Any spell of 3rd level or lower on the target ends" You can certainly target the familiar or the synaptically staticked creature but, as there are no active spells on them, nothing happens and you wasted your casting.

This was covered in Sage Advice. Link: https://www.dndbeyond.com/sources/sac/sage-advice-compendium#SA180

Can you use dispel magic on the creations of a spell like animate dead or affect those creations with antimagic field?

Whenever you wonder whether a spell’s effects can be dispelled or suspended, you need to answer one question: is the spell’s duration instantaneous? If the answer is yes, there is nothing to dispel or suspend. Here’s why: the effects of an instantaneous spell are brought into being by magic, but the effects aren’t sustained by magic. The magic flares for a split second and then vanishes. For example, the instantaneous spell animate dead harnesses magical energy to turn a corpse or a pile of bones into an undead creature. That necromantic magic is present for an instant and is then gone. The resulting undead now exists without the magic’s help. Casting dispel magic on the creature can’t end its mockery of life, and the undead can wander into an antimagic field with no adverse effect.

Another example: cure wounds instantaneously restores hit points to a creature. Because the spell’s duration is instantaneous, the restoration can’t be later dispelled. And you don’t suddenly lose hit points if you step into an antimagic field !

In contrast, a spell like conjure woodland beings has a non-instantaneous duration, which means its creations can be ended by dispel magic and they temporarily disappear within an antimagic field .

Quote from Cyb3rM1nd>>

There's multiple things that result in lasting magical effects even after the spell that caused the effects has ended. Find Familiar, is a great example. The familiar is magical and your bond is magical and you have the magical ability to store it and this is all caused by a spell. Yet, it is no longer a spell. It cannot be dispelled. 

Likewise Synaptic Static leaves a lasting magical effect in the mind of your target, but the spell that caused it has ended so this effect cannot be dispelled. 

Dispel Magic only works on active spells. You may target "magical effects" but that is only for targeting. Dispel only ends spells on that target. "Any spell of 3rd level or lower on the target ends" You can certainly target the familiar or the synaptically staticked creature but, as there are no active spells on them, nothing happens and you wasted your casting.

This was covered in Sage Advice. Link: https://www.dndbeyond.com/sources/sac/sage-advice-compendium#SA180

 To be fair, the Sage Advice you're quoting seems to contradict your thesis. "Here’s why: the effects of an instantaneous spell are brought into being by magic, but the effects aren’t sustained by magic. The magic flares for a split second and then vanishes." absolutely seems to suggest that any lingering effects are not magical. The magic has vanished.

My point is that Dispel Magic would not work.