Fill a large pot with lightly salted water and bring to a rolling boil over high heat. Stir in the fusilli; return to a boil.
Cook the pasta uncovered, until the pasta has cooked through, but is still firm to the bite, about 12 minutes.
Drain well and rinse with cold water.
Transfer cooled pasta to a large bowl.
Toss the carrot, onion, cucumber, celery, and salmon with the pasta; mix well.
Combine the mayonnaise, lemon juice, salt, pepper, and cayenne pepper in a small bowl; mix well.
Pour sauce over pasta mixture. Mix well to coat evenly.